# Quadratic Equations and Functions

## Solve Quadratic Equations Using the Square Root Property

A quadratic equation is an equation of the form ax2 + bx + c = 0, where . Quadratic equations differ from linear equations by including a quadratic term with the variable raised to the second power of the form ax2. We use different methods to solve quadratic equations than linear equations, because just adding, subtracting, multiplying, and dividing terms will not isolate the variable.
We have seen that some quadratic equations can be solved by factoring. In this chapter, we will learn three other methods to use in case a quadratic equation cannot be factored.

### Solve Quadratic Equations of the form  using the Square Root Property

We have already solved some quadratic equations by factoring. Let’s review how we used factoring to solve the quadratic equation x2 = 9.
We can easily use factoring to find the solutions of similar equations, like x2 = 16 and x2 = 25, because 16 and 25 are perfect squares. In each case, we would get two solutions,  and
But what happens when we have an equation like x2 = 7? Since 7 is not a perfect square, we cannot solve the equation by factoring.
Previously we learned that since 169 is the square of 13, we can also say that 13 is a square root of 169. Also, (−13)2 = 169, so −13 is also a square root of 169. Therefore, both 13 and −13 are square roots of 169. So, every positive number has two square roots—one positive and one negative. We earlier defined the square root of a number in this way:
Since these equations are all of the form x2 = k, the square root definition tells us the solutions are the two square roots of k. This leads to the Square Root Property.
Notice that the Square Root Property gives two solutions to an equation of the form x2 = k, the principal square root of  and its opposite. We could also write the solution as  We read this as x equals positive or negative the square root of k.
Now we will solve the equation x2 = 9 again, this time using the Square Root Property.
What happens when the constant is not a perfect square? Let’s use the Square Root Property to solve the equation x2 = 7.
We cannot simplify , so we leave the answer as a radical.
The steps to take to use the Square Root Property to solve a quadratic equation are listed here.
In order to use the Square Root Property, the coefficient of the variable term must equal one. In the next example, we must divide both sides of the equation by the coefficient 3 before using the Square Root Property.
The Square Root Property states ‘If ,’ What will happen if  This will be the case in the next example.
Our method also works when fractions occur in the equation; we solve as any equation with fractions. In the next example, we first isolate the quadratic term, and then make the coefficient equal to one.
The solutions to some equations may have fractions inside the radicals. When this happens, we must rationalize the denominator.

### Solve Quadratic Equations of the Form a(x − h)2 = k Using the Square Root Property

We can use the Square Root Property to solve an equation of the form a(x − h)2 = k as well. Notice that the quadratic term, x, in the original form ax2 = k is replaced with (x − h).
The first step, like before, is to isolate the term that has the variable squared. In this case, a binomial is being squared. Once the binomial is isolated, by dividing each side by the coefficient of a, then the Square Root Property can be used on (x − h)2.
Remember when we take the square root of a fraction, we can take the square root of the numerator and denominator separately.
We will start the solution to the next example by isolating the binomial term.
Sometimes the solutions are complex numbers.
The left sides of the equations in the next two examples do not seem to be of the form a(x − h)2. But they are perfect square trinomials, so we will factor to put them in the form we need.

### Key Concepts

1. Square Root Property

How to solve a quadratic equation using the square root property.

### Practice Makes Perfect

Solve Quadratic Equations of the Form
In the following exercises, solve each equation.
Solve Quadratic Equations of the Form
In the following exercises, solve each equation.

### Mixed Practice

In the following exercises, solve using the Square Root Property.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
Choose how would you respond to the statement “I can solve quadratic equations of the form a times the square of x minus h equals k using the Square Root Property.” “Confidently,” “with some help,” or “No, I don’t get it.”
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


# Quadratic Equations and Functions

## Solve Quadratic Equations by Completing the Square

So far we have solved quadratic equations by factoring and using the Square Root Property. In this section, we will solve quadratic equations by a process called completing the square, which is important for our work on conics later.

### Complete the Square of a Binomial Expression

In the last section, we were able to use the Square Root Property to solve the equation (y − 7)2 = 12 because the left side was a perfect square.
We also solved an equation in which the left side was a perfect square trinomial, but we had to rewrite it the form  in order to use the Square Root Property.
What happens if the variable is not part of a perfect square? Can we use algebra to make a perfect square?
Let’s look at two examples to help us recognize the patterns.
We restate the patterns here for reference.
We can use this pattern to “make” a perfect square.
We will start with the expression x2 + 6x. Since there is a plus sign between the two terms, we will use the (a + b)2 pattern, a2 + 2ab + b2 = (a + b)2.
We ultimately need to find the last term of this trinomial that will make it a perfect square trinomial. To do that we will need to find b. But first we start with determining a. Notice that the first term of x2 + 6x is a square, x2. This tells us that a = x.
What number, b, when multiplied with 2x gives 6x? It would have to be 3, which is  So b = 3.
Now to complete the perfect square trinomial, we will find the last term by squaring b, which is 32 = 9.
We can now factor.
So we found that adding 9 to x2 + 6x ‘completes the square’, and we write it as (x + 3)2.

### Solve Quadratic Equations of the Form x2 + bx + c = 0 by Completing the Square

In solving equations, we must always do the same thing to both sides of the equation. This is true, of course, when we solve a quadratic equation by completing the square too. When we add a term to one side of the equation to make a perfect square trinomial, we must also add the same term to the other side of the equation.
For example, if we start with the equation x2 + 6x = 40, and we want to complete the square on the left, we will add 9 to both sides of the equation.
Now the equation is in the form to solve using the Square Root Property! Completing the square is a way to transform an equation into the form we need to be able to use the Square Root Property.
The steps to solve a quadratic equation by completing the square are listed here.
When we solve an equation by completing the square, the answers will not always be integers.
In the previous example, our solutions were complex numbers. In the next example, the solutions will be irrational numbers.
We will start the next example by isolating the variable terms on the left side of the equation.
To solve the next equation, we must first collect all the variable terms on the left side of the equation. Then we proceed as we did in the previous examples.
Notice that the left side of the next equation is in factored form. But the right side is not zero. So, we cannot use the Zero Product Property since it says “If  then a = 0 or b = 0.” Instead, we multiply the factors and then put the equation into standard form to solve by completing the square.

### Solve Quadratic Equations of the Form ax2 + bx + c = 0 by Completing the Square

The process of completing the square works best when the coefficient of x2 is 1, so the left side of the equation is of the form x2 + bx + c. If the x2 term has a coefficient other than 1, we take some preliminary steps to make the coefficient equal to 1.
Sometimes the coefficient can be factored from all three terms of the trinomial. This will be our strategy in the next example.
To complete the square, the coefficient of the x2 must be 1. When the leading coefficient is not a factor of all the terms, we will divide both sides of the equation by the leading coefficient! This will give us a fraction for the second coefficient. We have already seen how to complete the square with fractions in this section.
Now that we have seen that the coefficient of x2 must be 1 for us to complete the square, we update our procedure for solving a quadratic equation by completing the square to include equations of the form ax2 + bx + c = 0.

### Key Concepts

1. Binomial Squares Pattern
If a and b are real numbers,

2. How to Complete a Square

3. How to solve a quadratic equation of the form ax2 + bx + c = 0 by completing the square.

### Practice Makes Perfect

Complete the Square of a Binomial Expression
In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.
Solve Quadratic Equations of the form
In the following exercises, solve by completing the square.
Solve Quadratic Equations of the form
In the following exercises, solve by completing the square.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Quadratic Equations and Functions

## Solve Quadratic Equations Using the Quadratic Formula

### Solve Quadratic Equations Using the Quadratic Formula

When we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier way to do this?’ The answer is ‘yes’. Mathematicians look for patterns when they do things over and over in order to make their work easier. In this section we will derive and use a formula to find the solution of a quadratic equation.
We have already seen how to solve a formula for a specific variable ‘in general’, so that we would do the algebraic steps only once, and then use the new formula to find the value of the specific variable. Now we will go through the steps of completing the square using the general form of a quadratic equation to solve a quadratic equation for x.
We start with the standard form of a quadratic equation and solve it for x by completing the square.
To use the Quadratic Formula, we substitute the values of a, b, and c from the standard form into the expression on the right side of the formula. Then we simplify the expression. The result is the pair of solutions to the quadratic equation.
Notice the formula is an equation. Make sure you use both sides of the equation.
If you say the formula as you write it in each problem, you’ll have it memorized in no time! And remember, the Quadratic Formula is an EQUATION. Be sure you start with “x =”.
When we solved quadratic equations by using the Square Root Property, we sometimes got answers that had radicals. That can happen, too, when using the Quadratic Formula. If we get a radical as a solution, the final answer must have the radical in its simplified form.
When we substitute a, b, and c into the Quadratic Formula and the radicand is negative, the quadratic equation will have imaginary or complex solutions. We will see this in the next example.
Remember, to use the Quadratic Formula, the equation must be written in standard form, ax2 + bx + c = 0. Sometimes, we will need to do some algebra to get the equation into standard form before we can use the Quadratic Formula.
When we solved linear equations, if an equation had too many fractions we cleared the fractions by multiplying both sides of the equation by the LCD. This gave us an equivalent equation—without fractions— to solve. We can use the same strategy with quadratic equations.
Think about the equation (x − 3)2 = 0. We know from the Zero Product Property that this equation has only one solution,x = 3.
We will see in the next example how using the Quadratic Formula to solve an equation whose standard form is a perfect square trinomial equal to 0 gives just one solution. Notice that once the radicand is simplified it becomes 0 , which leads to only one solution.

### Use the Discriminant to Predict the Number and Type of Solutions of a Quadratic Equation

When we solved the quadratic equations in the previous examples, sometimes we got two real solutions, one real solution, and sometimes two complex solutions. Is there a way to predict the number and type of solutions to a quadratic equation without actually solving the equation?
Yes, the expression under the radical of the Quadratic Formula makes it easy for us to determine the number and type of solutions. This expression is called the discriminant.
Let’s look at the discriminant of the equations in some of the examples and the number and type of solutions to those quadratic equations.

### Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

We summarize the four methods that we have used to solve quadratic equations below.
Given that we have four methods to use to solve a quadratic equation, how do you decide which one to use? Factoring is often the quickest method and so we try it first. If the equation is  or  we use the Square Root Property. For any other equation, it is probably best to use the Quadratic Formula.  Remember, you can solve any quadratic equation by using the Quadratic Formula, but that is not always the easiest method.
What about the method of Completing the Square? Most people find that method cumbersome and prefer not to use it. We needed to include it in the list of methods because we completed the square in general to derive the Quadratic Formula. You will also use the process of Completing the Square in other areas of algebra.
The next example uses this strategy to decide how to solve each quadratic equation.

### Key Concepts

1. Quadratic Formula

2. How to solve a quadratic equation using the Quadratic Formula.

3. Using the Discriminant, b2 − 4ac, to Determine the Number and Type of Solutions of a Quadratic Equation

4. Methods to Solve Quadratic Equations:

5. How to identify the most appropriate method to solve a quadratic equation.

### Practice Makes Perfect

Solve Quadratic Equations Using the Quadratic Formula
In the following exercises, solve by using the Quadratic Formula.
Use the Discriminant to Predict the Number of Real Solutions of a Quadratic Equation
In the following exercises, determine the number of real solutions for each quadratic equation.
Identify the Most Appropriate Method to Use to Solve a Quadratic Equation
In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


# Quadratic Equations and Functions

## Solve Equations in Quadratic Form

### Solve Equations in Quadratic Form

Sometimes when we factored trinomials, the trinomial did not appear to be in the ax2 + bx + c form. So we factored by substitution allowing us to make it fit the ax2 + bx + c form. We used the standard  for the substitution.
To factor the expression x4 − 4x2 − 5, we noticed the variable part of the middle term is x2 and its square, x4, is the variable part of the first term. (We know ) So we let u = x2 and factored.
Similarly, sometimes an equation is not in the ax2 + bx + c = 0 form but looks much like a quadratic equation. Then, we can often make a thoughtful substitution that will allow us to make it fit the ax2 + bx + c = 0 form. If we can make it fit the form, we can then use all of our methods to solve quadratic equations.
Notice that in the quadratic equation ax2 + bx + c = 0, the middle term has a variable, x, and its square, x2, is the variable part of the first term. Look for this relationship as you try to find a substitution.
Again, we will use the standard u to make a substitution that will put the equation in quadratic form. If the substitution gives us an equation of the form ax2 + bx + c = 0, we say the original equation was of quadratic form.
The next example shows the steps for solving an equation in quadratic form.
We summarize the steps to solve an equation in quadratic form.
In the next example, the binomial in the middle term, (x − 2) is squared in the first term. If we let u = x − 2 and substitute, our trinomial will be in ax2 + bx + c form.
In the next example, we notice that  Also, remember that when we square both sides of an equation, we may introduce extraneous roots. Be sure to check your answers!
Substitutions for rational exponents can also help us solve an equation in quadratic form. Think of the properties of exponents as you begin the next example.
In the next example, we need to keep in mind the definition of a negative exponent as well as the properties of exponents.

### Key Concepts

1. How to solve equations in quadratic form.

### Practice Makes Perfect

Solve Equations in Quadratic Form
In the following exercises, solve.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


# Quadratic Equations and Functions

## Solve Applications of Quadratic Equations

### Solve Applications Modeled by Quadratic Equations

We solved some applications that are modeled by quadratic equations earlier, when the only method we had to solve them was factoring. Now that we have more methods to solve quadratic equations, we will take another look at applications.
Let’s first summarize the methods we now have to solve quadratic equations.
As you solve each equation, choose the method that is most convenient for you to work the problem. As a reminder, we will copy our usual Problem-Solving Strategy here so we can follow the steps.
We have solved number applications that involved consecutive even and odd integers, by modeling the situation with linear equations. Remember, we noticed each even integer is 2 more than the number preceding it. If we call the first one n, then the next one is n + 2. The next one would be n + 2 + 2 or n + 4. This is also true when we use odd integers. One set of even integers and one set of odd integers are shown below.
Some applications of odd or even consecutive integers are modeled by quadratic equations. The notation above will be helpful as you name the variables.
We will use the formula for the area of a triangle to solve the next example.
Recall that when we solve geometric applications, it is helpful to draw the figure.
In the two preceding examples, the number in the radical in the Quadratic Formula was a perfect square and so the solutions were rational numbers. If we get an irrational number as a solution to an application problem, we will use a calculator to get an approximate value.
We will use the formula for the area of a rectangle to solve the next example.
The Pythagorean Theorem gives the relation between the legs and hypotenuse of a right triangle. We will use the Pythagorean Theorem to solve the next example.
The height of a projectile shot upward from the ground is modeled by a quadratic equation. The initial velocity, v0, propels the object up until gravity causes the object to fall back down.
We can use this formula to find how many seconds it will take for a firework to reach a specific height.
We have solved uniform motion problems using the formula D = rt in previous chapters. We used a table like the one below to organize the information and lead us to the equation.
The formula D = rt assumes we know r and t and use them to find D. If we know D and r and need to find t, we would solve the equation for t and get the formula
Some uniform motion problems are also modeled by quadratic equations.
Work applications can also be modeled by quadratic equations. We will set them up using the same methods we used when we solved them with rational equations.We’ll use a similar scenario now.

### Key Concepts

1. Methods to Solve Quadratic Equations

2. How to use a Problem-Solving Strategy.

3. Area of a Triangle

4. Area of a Rectangle

5. Pythagorean Theorem

6. Projectile motion

### Practice Makes Pefect

Solve Applications Modeled by Quadratic Equations
In the following exercises, solve using any method.
In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?


# Quadratic Equations and Functions

## Graph Quadratic Functions Using Properties

### Recognize the Graph of a Quadratic Function

Previously we very briefly looked at the function , which we called the square function. It was one of the first non-linear functions we looked at. Now we will graph functions of the form  if  We call this kind of function a quadratic function.
We graphed the quadratic function  by plotting points.
Every quadratic function has a graph that looks like this. We call this figure a parabola.
Let’s practice graphing a parabola by plotting a few points.
All graphs of quadratic functions of the form f (x) = ax2 + bx + c are parabolas that open upward or downward. See .
Notice that the only difference in the two functions is the negative sign before the quadratic term (x2 in the equation of the graph in ). When the quadratic term, is positive, the parabola opens upward, and when the quadratic term is negative, the parabola opens downward.

### Find the Axis of Symmetry and Vertex of a Parabola

Look again at . Do you see that we could fold each parabola in half and then one side would lie on top of the other? The ‘fold line’ is a line of symmetry. We call it the axis of symmetry of the parabola.
We show the same two graphs again with the axis of symmetry. See .
The equation of the axis of symmetry can be derived by using the Quadratic Formula. We will omit the derivation here and proceed directly to using the result. The equation of the axis of symmetry of the graph of f (x) = ax2 + bx + c is
So to find the equation of symmetry of each of the parabolas we graphed above, we will substitute into the formula
Notice that these are the equations of the dashed blue lines on the graphs.
The point on the parabola that is the lowest (parabola opens up), or the highest (parabola opens down), lies on the axis of symmetry. This point is called the vertex of the parabola.
We can easily find the coordinates of the vertex, because we know it is on the axis of symmetry. This means itsx-coordinate is  To find the y-coordinate of the vertex we substitute the value of the x-coordinate into the quadratic function.

### Find the Intercepts of a Parabola

When we graphed linear equations, we often used the x- and y-intercepts to help us graph the lines. Finding the coordinates of the intercepts will help us to graph parabolas, too.
Remember, at the y-intercept the value of x is zero. So to find the y-intercept, we substitute x = 0 into the function.
Let’s find the y-intercepts of the two parabolas shown in .
An x-intercept results when the value of f (x) is zero. To find an x-intercept, we let f (x) = 0. In other words, we will need to solve the equation 0 = ax2 + bx + c for x.
Solving quadratic equations like this is exactly what we have done earlier in this chapter!
We can now find the x-intercepts of the two parabolas we looked at. First we will find the x-intercepts of the parabola whose function is f (x) = x2 + 4x + 3.

Now we will find the x-intercepts of the parabola whose function is f (x) = −x2 + 4x + 3.

We will use the decimal approximations of the x-intercepts, so that we can locate these points on the graph,
Do these results agree with our graphs? See .
In this chapter, we have been solving quadratic equations of the form ax2 + bx + c = 0. We solved for x and the results were the solutions to the equation.
We are now looking at quadratic functions of the form f (x) = ax2 + bx + c. The graphs of these functions are parabolas. The x-intercepts of the parabolas occur where f (x) = 0.
For example:
The solutions of the quadratic function are the x values of the x-intercepts.
Earlier, we saw that quadratic equations have 2, 1, or 0 solutions. The graphs below show examples of parabolas for these three cases. Since the solutions of the functions give the x-intercepts of the graphs, the number of x-intercepts is the same as the number of solutions.
Previously, we used the discriminant to determine the number of solutions of a quadratic function of the form  Now we can use the discriminant to tell us how many x-intercepts there are on the graph.
Before you to find the values of the x-intercepts, you may want to evaluate the discriminant so you know how many solutions to expect.

### Graph Quadratic Functions Using Properties

Now we have all the pieces we need in order to graph a quadratic function. We just need to put them together. In the next example we will see how to do this.
We list the steps to take in order to graph a quadratic function here.
We were able to find the x-intercepts in the last example by factoring. We find the x-intercepts in the next example by factoring, too.
For the graph of f (x) = −x2 + 6x − 9, the vertex and the x-intercept were the same point. Remember how the discriminant determines the number of solutions of a quadratic equation? The discriminant of the equation 0 = −x2 + 6x − 9 is 0, so there is only one solution. That means there is only one x-intercept, and it is the vertex of the parabola.
How many x-intercepts would you expect to see on the graph of f (x) = x2 + 4x + 5?
Finding the y-intercept by finding f (0) is easy, isn’t it? Sometimes we need to use the Quadratic Formula to find the x-intercepts.

### Solve Maximum and Minimum Applications

Knowing that the vertex of a parabola is the lowest or highest point of the parabola gives us an easy way to determine the minimum or maximum value of a quadratic function. The y-coordinate of the vertex is the minimum value of a parabola that opens upward. It is the maximum value of a parabola that opens downward. See .
We have used the formula
to calculate the height in feet, h , of an object shot upwards into the air with initial velocity, v0, after t seconds .
This formula is a quadratic function, so its graph is a parabola. By solving for the coordinates of the vertex (t, h), we can find how long it will take the object to reach its maximum height. Then we can calculate the maximum height.

### Key Concepts

1. Parabola Orientation

2. Axis of Symmetry and Vertex of a Parabola The graph of the function  is a parabola where:

3. Find the Intercepts of a Parabola

4. How to graph a quadratic function using properties.

5. Minimum or Maximum Values of a Quadratic Equation

### Practice Makes Perfect

Recognize the Graph of a Quadratic Function
In the following exercises, graph the functions by plotting points.
For each of the following exercises, determine if the parabola opens up or down.
Find the Axis of Symmetry and Vertex of a Parabola
In the following functions, find ⓐ the equation of the axis of symmetry and ⓑ the vertex of its graph.
Find the Intercepts of a Parabola
In the following exercises, find the intercepts of the parabola whose function is given.
Graph Quadratic Functions Using Properties
In the following exercises, graph the function by using its properties.
Solve Maximum and Minimum Applications
In the following exercises, find the maximum or minimum value of each function.
In the following exercises, solve. Round answers to the nearest tenth.

### Writing Exercise

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?


# Quadratic Equations and Functions

## Graph Quadratic Functions Using Transformations

### Graph Quadratic Functions of the form

In the last section, we learned how to graph quadratic functions using their properties. Another method involves starting with the basic graph of  and ‘moving’ it according to information given in the function equation. We call this graphing quadratic functions using transformations.
In the first example, we will graph the quadratic function  by plotting points. Then we will see what effect adding a constant, k, to the equation will have on the graph of the new function
The last example shows us that to graph a quadratic function of the form  we take the basic parabola graph of  and vertically shift it up  or shift it down .
This transformation is called a vertical shift.
Now that we have seen the effect of the constant, k, it is easy to graph functions of the form  We just start with the basic parabola of  and then shift it up or down.
It may be helpful to practice sketching  quickly. We know the values and can sketch the graph from there.
Once we know this parabola, it will be easy to apply the transformations. The next example will require a vertical shift.

### Graph Quadratic Functions of the form

In the first example, we graphed the quadratic function  by plotting points and then saw the effect of adding a constant k to the function had on the resulting graph of the new function
We will now explore the effect of subtracting a constant, h, from x has on the resulting graph of the new function
The last example shows us that to graph a quadratic function of the form  we take the basic parabola graph of  and shift it left (h > 0) or shift it right (h < 0).
This transformation is called a horizontal shift.
Now that we have seen the effect of the constant, h, it is easy to graph functions of the form  We just start with the basic parabola of  and then shift it left or right.
The next example will require a horizontal shift.
Now that we know the effect of the constants h and k, we will graph a quadratic function of the form  by first drawing the basic parabola and then making a horizontal shift followed by a vertical shift. We could do the vertical shift followed by the horizontal shift, but most students prefer the horizontal shift followed by the vertical.

### Graph Quadratic Functions of the Form

So far we graphed the quadratic function  and then saw the effect of including a constant h or k in the equation had on the resulting graph of the new function. We will now explore the effect of the coefficient a on the resulting graph of the new function
If we graph these functions, we can see the effect of the constant a, assuming a > 0.
To graph a function with constant a it is easiest to choose a few points on  and multiply the y-values by a.

### Graph Quadratic Functions Using Transformations

We have learned how the constants a, h, and k in the functions,  and  affect their graphs. We can now put this together and graph quadratic functions  by first putting them into the form  by completing the square. This form is sometimes known as the vertex form or standard form.
We must be careful to both add and subtract the number to the SAME side of the function to complete the square. We cannot add the number to both sides as we did when we completed the square with quadratic equations.
When we complete the square in a function with a coefficient of x2 that is not one, we have to factor that coefficient from just the x-terms. We do not factor it from the constant term. It is often helpful to move the constant term a bit to the right to make it easier to focus only on the x-terms.
Once we get the constant we want to complete the square, we must remember to multiply it by that coefficient before we then subtract it.
Once we put the function into the  form, we can then use the transformations as we did in the last few problems. The next example will show us how to do this.
We list the steps to take to graph a quadratic function using transformations here.
Now that we have completed the square to put a quadratic function into  form, we can also use this technique to graph the function using its properties as in the previous section.
If we look back at the last few examples, we see that the vertex is related to the constants h and k.
In each case, the vertex is (h, k). Also the axis of symmetry is the line x = h.
We rewrite our steps for graphing a quadratic function using properties for when the function is in  form.

### Find a Quadratic Function from its Graph

So far we have started with a function and then found its graph.
Now we are going to reverse the process. Starting with the graph, we will find the function.

### Key Concepts

1. Graph a Quadratic Function of the form  Using a Vertical Shift

2. Graph a Quadratic Function of the form  Using a Horizontal Shift

3. Graph of a Quadratic Function of the form

4. How to graph a quadratic function using transformations

5. Graph a quadratic function in the vertex form  using properties

### Practice Makes Perfect

Graph Quadratic Functions of the form
In the following exercises, ⓐ graph the quadratic functions on the same rectangular coordinate system and ⓑ describe what effect adding a constant, k, to the function has on the basic parabola.
In the following exercises, graph each function using a vertical shift.
Graph Quadratic Functions of the form
In the following exercises, ⓐ graph the quadratic functions on the same rectangular coordinate system and ⓑ describe what effect adding a constant, , inside the parentheses has
In the following exercises, graph each function using a horizontal shift.
In the following exercises, graph each function using transformations.
Graph Quadratic Functions of the form
In the following exercises, graph each function.
Graph Quadratic Functions Using Transformations
In the following exercises, rewrite each function in the  form by completing the square.
In the following exercises, ⓐ rewrite each function in  form and ⓑ graph it by using transformations.
In the following exercises, ⓐ rewrite each function in  form and ⓑ graph it using properties.
Matching
In the following exercises, match the graphs to one of the following functions: ⓐ  ⓑ  ⓒ  ⓓ  ⓔ  ⓕ  ⓖ  ⓗ
Find a Quadratic Function from its Graph
In the following exercises, write the quadratic function in  form whose graph is shown.

### Writing Exercise

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?


# Quadratic Equations and Functions

## Solve Quadratic Inequalities

We have learned how to solve linear inequalities and rational inequalities previously. Some of the techniques we used to solve them were the same and some were different.
We will now learn to solve inequalities that have a quadratic expression. We will use some of the techniques from solving linear and rational inequalities as well as quadratic equations.
We will solve quadratic inequalities two ways—both graphically and algebraically.

### Solve Quadratic Inequalities Graphically

A quadratic equation is in standard form when written as ax2 + bx + c = 0. If we replace the equal sign with an inequality sign, we have a quadratic inequality in standard form.
The graph of a quadratic function f(x) = ax2 + bx + c = 0 is a parabola. When we ask when is ax2 + bx + c < 0, we are asking when is f(x) < 0. We want to know when the parabola is below the x-axis.
When we ask when is ax2 + bx + c > 0, we are asking when is f(x) > 0. We want to know when the parabola is above the x-axis.
We list the steps to take to solve a quadratic inequality graphically.
In the last example, the parabola opened upward and in the next example, it opens downward. In both cases, we are looking for the part of the parabola that is below the x-axis but note how the position of the parabola affects the solution.

### Solve Quadratic Inequalities Algebraically

The algebraic method we will use is very similar to the method we used to solve rational inequalities. We will find the critical points for the inequality, which will be the solutions to the related quadratic equation. Remember a polynomial expression can change signs only where the expression is zero.
We will use the critical points to divide the number line into intervals and then determine whether the quadratic expression willl be postive or negative in the interval. We then determine the solution for the inequality.
In this example, since the expression  factors nicely, we can also find the sign in each interval much like we did when we solved rational inequalities. We find the sign of each of the factors, and then the sign of the product. Our number line would like this:
The result is the same as we found using the other method.
We summarize the steps here.
The solutions of the quadratic inequalities in each of the previous examples, were either an interval or the union of two intervals. This resulted from the fact that, in each case we found two solutions to the corresponding quadratic equation ax2 + bx + c = 0. These two solutions then gave us either the two x-intercepts for the graph or the two critical points to divide the number line into intervals.
This correlates to our previous discussion of the number and type of solutions to a quadratic equation using the discriminant.
For a quadratic equation of the form ax2 + bx + c = 0,
The last row of the table shows us when the parabolas never intersect the x-axis. Using the Quadratic Formula to solve the quadratic equation, the radicand is a negative. We get two complex solutions.
In the next example, the quadratic inequality solutions will result from the solution of the quadratic equation being complex.

### Key Concepts

1. Solve a Quadratic Inequality Graphically

2. How to Solve a Quadratic Inequality Algebraically

### Section Exercises

### Practice Makes Perfect

Solve Quadratic Inequalities Graphically
In the following exercises, ⓐ solve graphically and ⓑ write the solution in interval notation.
In the following exercises, solve each inequality algebraically and write any solution in interval notation.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

### Chapter Review Exercises

### Solve Quadratic Equations Using the Square Root Property

Solve Quadratic Equations of the form
In the following exercises, solve using the Square Root Property.
Solve Quadratic Equations of the Form
In the following exercises, solve using the Square Root Property.

### Solve Quadratic Equations by Completing the Square

Solve Quadratic Equations Using Completing the Square
In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.
In the following exercises, solve by completing the square.
Solve Quadratic Equations of the form
In the following exercises, solve by completing the square.

### Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.
Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation
In the following exercises, determine the number of solutions for each quadratic equation.
Identify the Most Appropriate Method to Use to Solve a Quadratic Equation
In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

### Solve Equations in Quadratic Form

Solve Equations in Quadratic Form
In the following exercises, solve.

### Solve Applications of Quadratic Equations

Solve Applications Modeled by Quadratic Equations
In the following exercises, solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

### Graph Quadratic Functions Using Properties

Recognize the Graph of a Quadratic Function
In the following exercises, graph by plotting point.
In the following exercises, determine if the following parabolas open up or down.
Find the Axis of Symmetry and Vertex of a Parabola
In the following exercises, find ⓐ the equation of the axis of symmetry and ⓑ the vertex.
Find the Intercepts of a Parabola
In the following exercises, find the x- and y-intercepts.
Graph Quadratic Functions Using Properties
In the following exercises, graph by using its properties.
Solve Maximum and Minimum Applications
In the following exercises, find the minimum or maximum value.
In the following exercises, solve. Rounding answers to the nearest tenth.

### Graph Quadratic Functions Using Transformations

Graph Quadratic Functions of the form
In the following exercises, graph each function using a vertical shift.
In the following exercises, graph each function using a horizontal shift.
In the following exercises, graph each function using transformations.
Graph Quadratic Functions of the form
In the following exercises, graph each function.
Graph Quadratic Functions Using Transformations
In the following exercises, rewrite each function in the  form by completing the square.
In the following exercises, ⓐ rewrite each function in  form and ⓑ graph it by using transformations.
In the following exercises, ⓐ rewrite each function in  form and ⓑ graph it using properties.
Find a Quadratic Function from its Graph
In the following exercises, write the quadratic function in  form.

### Solve Quadratic Inequalities

Solve Quadratic Inequalities Graphically
In the following exercises, solve graphically and write the solution in interval notation.
In the following exercises, solve each inequality algebraically and write any solution in interval notation.

### Practice Test

Solve the following quadratic equations. Use any method.
Use the discriminant to determine the number and type of solutions of each quadratic equation.
Solve each equation.
For each parabola, find ⓐ which direction it opens, ⓑ the equation of the axis of symmetry, ⓒ the vertex, ⓓ the x- and y-intercepts, and e) the maximum or minimum value.
Graph each quadratic function using intercepts, the vertex, and the equation of the axis of symmetry.
In the following exercises, graph each function using transformations.
In the following exercises, solve each inequality algebraically and write any solution in interval notation.
Model the situation with a quadratic equation and solve by any method.


# Exponential and Logarithmic Functions

## Introduction

As the world population continues to grow, food supplies are becoming less able to meet the increasing demand. At the same time, available resources of fertile soil for growing plants is dwindling. One possible solution—grow plants without soil. Botanists around the world are expanding the potential of hydroponics, which is the process of growing plants without soil. To provide the plants with the nutrients they need, the botanists keep careful growth records. Some growth is described by the types of functions you will explore in this chapter—exponential and logarithmic. You will evaluate and graph these functions, and solve equations using them.


# Exponential and Logarithmic Functions

## Finding Composite and Inverse Functions

In this chapter, we will introduce two new types of functions, exponential functions and logarithmic functions. These functions are used extensively in business and the sciences as we will see.

### Find and Evaluate Composite Functions

Before we introduce the functions, we need to look at another operation on functions called composition. In composition, the output of one function is the input of a second function. For functions  and  the composition is written  and is defined by
We read  as  of  of
To do a composition, the output of the first function,  becomes the input of the second function, f, and so we must be sure that it is part of the domain of f.
We have actually used composition without using the notation many times before. When we graphed quadratic functions using translations, we were composing functions. For example, if we first graphed  as a parabola and then shifted it down vertically four units, we were using the composition defined by  where
The next example will demonstrate that   and  usually result in different outputs.
In the next example we will evaluate a composition for a specific value.

### Determine Whether a Function is One-to-One

When we first introduced functions, we said a function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each x-value is matched with only one y-value.
We used the birthday example to help us understand the definition. Every person has a birthday, but no one has two birthdays and it is okay for two people to share a birthday. Since each person has exactly one birthday, that relation is a function.
A function is one-to-one if each value in the range has exactly one element in the domain. For each ordered pair in the function, each y-value is matched with only one x-value.
Our example of the birthday relation is not a one-to-one function. Two people can share the same birthday. The range value August 2 is the birthday of Liz and June, and so one range value has two domain values. Therefore, the function is not one-to-one.
To help us determine whether a relation is a function, we use the vertical line test. A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. Also, if any vertical line intersects the graph in more than one point, the graph does not represent a function.
The vertical line is representing an x-value and we check that it intersects the graph in only one y-value. Then it is a function.
To check if a function is one-to-one, we use a similar process. We use a horizontal line and check that each horizontal line intersects the graph in only one point. The horizontal line is representing a y-value and we check that it intersects the graph in only one x-value. If every horizontal line intersects the graph of a function in at most one point, it is a one-to-one function. This is the horizontal line test.
We can test whether a graph of a relation is a function by using the vertical line test. We can then tell if the function is one-to-one by applying the horizontal line test.

### Find the Inverse of a Function

Let’s look at a one-to one function, , represented by the ordered pairs  For each -value,  adds 5 to get the -value. To ‘undo’ the addition of 5, we subtract 5 from each -value and get back to the original -value. We can call this “taking the inverse of ” and name the function
Notice that that the ordered pairs of  and  have their -values and -values reversed. The domain of  is the range of  and the domain of  is the range of
In the next example we will find the inverse of a function defined by ordered pairs.
We just noted that if  is a one-to-one function whose ordered pairs are of the form  then its inverse function  is the set of ordered pairs
So if a point  is on the graph of a function  then the ordered pair  is on the graph of  See .
The distance between any two pairs  and  is cut in half by the line  So we say the points are mirror images of each other through the line
Since every point on the graph of a function  is a mirror image of a point on the graph of  we say the graphs are mirror images of each other through the line  We will use this concept to graph the inverse of a function in the next example.
When we began our discussion of an inverse function, we talked about how the inverse function ‘undoes’ what the original function did to a value in its domain in order to get back to the original x-value.
We can use this property to verify that two functions are inverses of each other.
We have found inverses of function defined by ordered pairs and from a graph. We will now look at how to find an inverse using an algebraic equation. The method uses the idea that if  is a one-to-one function with ordered pairs  then its inverse function  is the set of ordered pairs
If we reverse the x and y in the function and then solve for y, we get our inverse function.
We summarize the steps below.

### Key Concepts

1. Composition of Functions: The composition of functions  and  is written  and is defined by

We read  as  of  of
2. Horizontal Line Test: If every horizontal line, intersects the graph of a function in at most one point, it is a one-to-one function.
3. Inverse of a Function Defined by Ordered Pairs: If  is a one-to-one function whose ordered pairs are of the form  then its inverse function  is the set of ordered pairs
4. Inverse Functions: For every  in the domain of one-to-one function  and

5. How to Find the Inverse of a One-to-One Function:

### Practice Makes Perfect

Find and Evaluate Composite Functions
In the following exercises, find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).
In the following exercises, find the values described.
Determine Whether a Function is One-to-One
In the following exercises, determine if the set of ordered pairs represents a function and if so, is the function one-to-one.
In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.
In the following exercises, find the inverse of each function. Determine the domain and range of the inverse function.
In the following exercises, graph, on the same coordinate system, the inverse of the one-to-one function shown.
In the following exercises, determine whether or not the given functions are inverses.
In the following exercises, find the inverse of each function.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


# Exponential and Logarithmic Functions

## Evaluate and Graph Exponential Functions

### Graph Exponential Functions

The functions we have studied so far do not give us a model for many naturally occurring phenomena. From the growth of populations and the spread of viruses to radioactive decay and compounding interest, the models are very different from what we have studied so far. These models involve exponential functions.
An exponential function is a function of the form  where  and
Notice that in this function, the variable is the exponent. In our functions so far, the variables were the base.
Our definition says  If we let  then  becomes  Since  for all real numbers,  This is the constant function.
Our definition also says  If we let a base be negative, say  then  is not a real number when
In fact,  would not be a real number any time  is a fraction with an even denominator. So our definition requires
By graphing a few exponential functions, we will be able to see their unique properties.
If we look at the graphs from the previous Example and Try Its, we can identify some of the properties of exponential functions.
The graphs of  and  as well as the graphs of  and  all have the same basic shape. This is the shape we expect from an exponential function where
We notice, that for each function, the graph contains the point  This make sense because  for any a.
The graph of each function,  also contains the point  The graph of  contained  and the graph of  contained  This makes sense as
Notice too, the graph of each function  also contains the point  The graph of  contained  and the graph of  contained  This makes sense as
What is the domain for each function? From the graphs we can see that the domain is the set of all real numbers. There is no restriction on the domain. We write the domain in interval notation as
Look at each graph. What is the range of the function? The graph never hits the -axis. The range is all positive numbers. We write the range in interval notation as
Whenever a graph of a function approaches a line but never touches it, we call that line an asymptote. For the exponential functions we are looking at, the graph approaches the -axis very closely but will never cross it, we call the line  the x-axis, a horizontal asymptote.
Our definition of an exponential function  says  but the examples and discussion so far has been about functions where  What happens when ? The next example will explore this possibility.
Now let’s look at the graphs from the previous Example and Try Its so we can now identify some of the properties of exponential functions where
The graphs of  and  as well as the graphs of  and  all have the same basic shape. While this is the shape we expect from an exponential function where  the graphs go down from left to right while the previous graphs, when  went from up from left to right.
We notice that for each function, the graph still contains the point (0, 1). This make sense because  for any a.
As before, the graph of each function,  also contains the point  The graph of  contained  and the graph of  contained  This makes sense as
Notice too that the graph of each function,  also contains the point  The graph of  contained  and the graph of  contained  This makes sense as
What is the domain and range for each function? From the graphs we can see that the domain is the set of all real numbers and we write the domain in interval notation as  Again, the graph never hits the -axis. The range is all positive numbers. We write the range in interval notation as
We will summarize these properties in the chart below. Which also include when
It is important for us to notice that both of these graphs are one-to-one, as they both pass the horizontal line test. This means the exponential function will have an inverse. We will look at this later.
When we graphed quadratic functions, we were able to graph using translation rather than just plotting points. Will that work in graphing exponential functions?
Looking at the graphs of the functions  and  in the last example, we see that adding one in the exponent caused a horizontal shift of one unit to the left. Recognizing this pattern allows us to graph other functions with the same pattern by translation.
Let’s now consider another situation that might be graphed more easily by translation, once we recognize the pattern.
Looking at the graphs of the functions  and  in the last example, we see that subtracting 2 caused a vertical shift of down two units. Notice that the horizontal asymptote also shifted down 2 units. Recognizing this pattern allows us to graph other functions with the same pattern by translation.
All of our exponential functions have had either an integer or a rational number as the base. We will now look at an exponential function with an irrational number as the base.
Before we can look at this exponential function, we need to define the irrational number, e. This number is used as a base in many applications in the sciences and business that are modeled by exponential functions. The number is defined as the value of  as n gets larger and larger. We say, as n approaches infinity, or increases without bound. The table shows the value of  for several values of
The number e is like the number  in that we use a symbol to represent it because its decimal representation never stops or repeats. The irrational number e is called the natural base.
The exponential function whose base is   is called the natural exponential function.
Let’s graph the function  on the same coordinate system as  and
Notice that the graph of  is “between” the graphs of  and  Does this make sense as ?

### Solve Exponential Equations

Equations that include an exponential expression  are called exponential equations. To solve them we use a property that says as long as  and  if  then it is true that  In other words, in an exponential equation, if the bases are equal then the exponents are equal.
To use this property, we must be certain that both sides of the equation are written with the same base.
The steps are summarized below.
In the next example, we will use our properties on exponents.

### Use Exponential Models in Applications

Exponential functions model many situations. If you own a bank account, you have experienced the use of an exponential function. There are two formulas that are used to determine the balance in the account when interest is earned. If a principal, P, is invested at an interest rate, r, for t years, the new balance, A, will depend on how often the interest is compounded. If the interest is compounded n times a year we use the formula  If the interest is compounded continuously, we use the formula  These are the formulas for compound interest.
As you work with the Interest formulas, it is often helpful to identify the values of the variables first and then substitute them into the formula.
Other topics that are modeled by exponential functions involve growth and decay. Both also use the formula  we used for the growth of money. For growth and decay, generally we use as the original amount instead of calling it  the principal. We see that exponential growth has a positive rate of growth and exponential decay has a negative rate of growth.
Exponential growth is typically seen in the growth of populations of humans or animals or bacteria. Our next example looks at the growth of a virus.

### Key Concepts

1. Properties of the Graph of

2. One-to-One Property of Exponential Functions:
For  and

3. How to Solve an Exponential Equation

4. Compound Interest: For a principal,  invested at an interest rate,  for  years, the new balance,  is

5. Exponential Growth and Decay: For an original amount,  that grows or decays at a rate,  for a certain time  the final amount, is

### Practice Makes Perfect

Graph Exponential Functions
In the following exercises, graph each exponential function.
In the following exercises, graph each function in the same coordinate system.
In the following exercises, graph each exponential function.
Solve Exponential Equations
In the following exercises, solve each equation.
In the following exercises, match the graphs to one of the following functions: ⓐ  ⓑ  ⓒ  ⓓ  ⓔ  ⓕ
Use exponential models in applications
In the following exercises, use an exponential model to solve.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Exponential and Logarithmic Functions

## Evaluate and Graph Logarithmic Functions

We have spent some time finding the inverse of many functions. It works well to ‘undo’ an operation with another operation. Subtracting ‘undoes’ addition, multiplication ‘undoes’ division, taking the square root ‘undoes’ squaring.
As we studied the exponential function, we saw that it is one-to-one as its graphs pass the horizontal line test. This means an exponential function does have an inverse. If we try our algebraic method for finding an inverse, we run into a problem.

To deal with this we define the logarithm function with base a to be the inverse of the exponential function  We use the notation  and say the inverse function of the exponential function is the logarithmic function.

### Convert Between Exponential and Logarithmic Form

Since the equations  and  are equivalent, we can go back and forth between them. This will often be the method to solve some exponential and logarithmic equations. To help with converting back and forth let’s take a close look at the equations. See . Notice the positions of the exponent and base.
If we realize the logarithm is the exponent it makes the conversion easier. You may want to repeat, “base to the exponent give us the number.”
In the next example we do the reverse—convert logarithmic form to exponential form.

### Evaluate Logarithmic Functions

We can solve and evaluate logarithmic equations by using the technique of converting the equation to its equivalent exponential equation.
When see an expression such as  we can find its exact value two ways. By inspection we realize it means  to what power will be  Since  we know  An alternate way is to set the expression equal to  and then convert it into an exponential equation.

### Graph Logarithmic Functions

To graph a logarithmic function  it is easiest to convert the equation to its exponential form,  Generally, when we look for ordered pairs for the graph of a function, we usually choose an x-value and then determine its corresponding y-value. In this case you may find it easier to choose y-values and then determine its corresponding x-value.
The graphs of  and  are the shape we expect from a logarithmic function where
We notice that for each function the graph contains the point  This make sense because  means  which is true for any a.
The graph of each function, also contains the point  This makes sense as  means  which is true for any a.
Notice too, the graph of each function  also contains the point  This makes sense as  means  which is true for any a.
Look at each graph again. Now we will see that many characteristics of the logarithm function are simply ’mirror images’ of the characteristics of the corresponding exponential function.
What is the domain of the function? The graph never hits the y-axis. The domain is all positive numbers. We write the domain in interval notation as
What is the range for each function? From the graphs we can see that the range is the set of all real numbers. There is no restriction on the range. We write the range in interval notation as
When the graph approaches the y-axis so very closely but will never cross it, we call the line  the y-axis, a vertical asymptote.
Our next example looks at the graph of  when
Now, let’s look at the graphs  and , so we can identify some of the properties of logarithmic functions where
The graphs of all have the same basic shape. While this is the shape we expect from a logarithmic function where
We notice, that for each function again, the graph contains the points, This make sense for the same reasons we argued above.
We notice the domain and range are also the same—the domain is  and the range is  The -axis is again the vertical asymptote.
We will summarize these properties in the chart below. Which also include when
We talked earlier about how the logarithmic function  is the inverse of the exponential function  The graphs in  show both the exponential (blue) and logarithmic (red) functions on the same graph for both  and
Notice how the graphs are reflections of each other through the line  We know this is true of inverse functions. Keeping a visual in your mind of these graphs will help you remember the domain and range of each function. Notice the x-axis is the horizontal asymptote for the exponential functions and the y-axis is the vertical asymptote for the logarithmic functions.

### Solve Logarithmic Equations

When we talked about exponential functions, we introduced the number e. Just as e was a base for an exponential function, it can be used a base for logarithmic functions too. The logarithmic function with base e is called the natural logarithmic function. The function  is generally written  and we read it as “el en of
When the base of the logarithm function is 10, we call it the common logarithmic function and the base is not shown. If the base a of a logarithm is not shown, we assume it is 10.
To solve logarithmic equations, one strategy is to change the equation to exponential form and then solve the exponential equation as we did before. As we solve logarithmic equations, , we need to remember that for the base a,  and  Also, the domain is  Just as with radical equations, we must check our solutions to eliminate any extraneous solutions.

### Use Logarithmic Models in Applications

There are many applications that are modeled by logarithmic equations. We will first look at the logarithmic equation that gives the decibel (dB) level of sound. Decibels range from 0, which is barely audible to 160, which can rupture an eardrum. The  in the formula represents the intensity of sound that is barely audible.
The magnitude  of an earthquake is measured by a logarithmic scale called the Richter scale. The model is  where  is the intensity of the shock wave. This model provides a way to measure earthquake intensity.

### Key Concepts

1. Properties of the Graph of

2. Decibel Level of Sound: The loudness level, , measured in decibels, of a sound of intensity, , measured in watts per square inch is
3. Earthquake Intensity: The magnitude  of an earthquake is measured by  where  is the intensity of its shock wave.

### Practice Makes Perfect

Convert Between Exponential and Logarithmic Form
In the following exercises, convert from exponential to logarithmic form.
In the following exercises, convert each logarithmic equation to exponential form.
Evaluate Logarithmic Functions
In the following exercises, find the value of  in each logarithmic equation.
In the following exercises, find the exact value of each logarithm without using a calculator.
Graph Logarithmic Functions
In the following exercises, graph each logarithmic function.
Solve Logarithmic Equations
In the following exercises, solve each logarithmic equation.
Use Logarithmic Models in Applications
In the following exercises, use a logarithmic model to solve.

### Writing Exercises

### Self Check

ⓐ  After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Exponential and Logarithmic Functions

## Use the Properties of Logarithms

### Use the Properties of Logarithms

Now that we have learned about exponential and logarithmic functions, we can introduce some of the properties of logarithms. These will be very helpful as we continue to solve both exponential and logarithmic equations.
The first two properties derive from the definition of logarithms. Since  we can convert this to logarithmic form and get  Also, since  we get
In the next example we could evaluate the logarithm by converting to exponential form, as we have done previously, but  recognizing and then applying the properties saves time.
The next two properties can also be verified by converting them from exponential form to logarithmic form, or the reverse.
The exponential equation  converts to the logarithmic equation  which is a true statement for positive values for x only.
The logarithmic equation  converts to the exponential equation  which is also a true statement.
These two properties are called inverse properties because, when we have the same base, raising to a power “undoes” the log and taking the log “undoes” raising to a power. These two properties show the composition of functions. Both ended up with the identity function which shows again that the exponential and logarithmic functions are inverse functions.
In the next example, apply the inverse properties of logarithms.
There are three more properties of logarithms that will be useful in our work. We know exponential functions and logarithmic function are very interrelated. Our definition of logarithm shows us that a logarithm is the exponent of the equivalent exponential. The properties of exponents have related properties for exponents.
In the Product Property of Exponents,  we see that to multiply the same base, we add the exponents. The Product Property of Logarithms,  tells us to take the log of a product, we add the log of the factors.
We use this property to write the log of a product as a sum of the logs of each factor.
Similarly, in the Quotient Property of Exponents,  we see that to divide the same base, we subtract the exponents. The Quotient Property of Logarithms,  tells us to take the log of a quotient, we subtract the log of the numerator and denominator.
Note that
We use this property to write the log of a quotient as a difference of the logs of each factor.
The third property of logarithms is related to the Power Property of Exponents,  we see that to raise a power to a power, we multiply the exponents. The Power Property of Logarithms,  tells us to take the log of a number raised to a power, we multiply the power times the log of the number.
We use this property to write the log of a number raised to a power as the product of the power times the log of the number. We essentially take the exponent and throw it in front of the logarithm.
We summarize the Properties of Logarithms here for easy reference. While the natural logarithms are a special case of these properties, it is often helpful to also show the natural logarithm version of each property.
Now that we have the properties we can use them to “expand” a logarithmic expression. This means to write the logarithm as a sum or difference and without any powers.
We generally apply the Product and Quotient Properties before we apply the Power Property.
When we have a radical in the logarithmic expression, it is helpful to first write its radicand as a rational exponent.
The opposite of expanding a logarithm is to condense a sum or difference of logarithms that have the same base into a single logarithm. We again use the properties of logarithms to help us, but in reverse.
To condense logarithmic expressions with the same base into one logarithm, we start by using the Power Property to get the coefficients of the log terms to be one and then the Product and Quotient Properties as needed.

### Use the Change-of-Base Formula

To evaluate a logarithm with any other base, we can use the Change-of-Base Formula. We will show how this is derived.
The Change-of-Base Formula introduces a new base  This can be any base b we want where  Because our calculators have keys for logarithms base 10 and base e, we will rewrite the Change-of-Base Formula with the new base as 10 or e.
When we use a calculator to find the logarithm value, we usually round to three decimal places. This gives us an approximate value and so we use the approximately equal symbol .

### Key Concepts

1. Properties of Logarithms

2. Inverse Properties of Logarithms

3. Product Property of Logarithms

4. Quotient Property of Logarithms

5. Power Property of Logarithms

6. Properties of Logarithms Summary
If  and  is any real number then,

7. Change-of-Base Formula
For any logarithmic bases a and b, and

### Practice Makes Perfect

Use the Properties of Logarithms
In the following exercises, use the properties of logarithms to evaluate.
In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible.
In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.
In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.
Use the Change-of-Base Formula
In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of  how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


# Exponential and Logarithmic Functions

## Solve Exponential and Logarithmic Equations

### Solve Logarithmic Equations Using the Properties of Logarithms

In the section on logarithmic functions, we solved some equations by rewriting the equation in exponential form. Now that we have the properties of logarithms, we have additional methods we can use to solve logarithmic equations.
If our equation has two logarithms we can use a property that says that if  then it is true that  This is the One-to-One Property of Logarithmic Equations.
To use this property, we must be certain that both sides of the equation are written with the same base.
Remember that logarithms are defined only for positive real numbers. Check your results in the original equation. You may have obtained a result that gives a logarithm of zero or a negative number.
Another strategy to use to solve logarithmic equations is to condense sums or differences into a single logarithm.
When there are logarithms on both sides, we condense each side into a single logarithm. Remember to use the Power Property as needed.

### Solve Exponential Equations Using Logarithms

In the section on exponential functions, we solved some equations by writing both sides of the equation with the same base. Next we wrote a new equation by setting the exponents equal.
It is not always possible or convenient to write the expressions with the same base. In that case we often take the common logarithm or natural logarithm of both sides once the exponential is isolated.
When we take the logarithm of both sides we will get the same result whether we use the common or the natural logarithm (try using the natural log in the last example. Did you get the same result?) When the exponential has base e, we use the natural logarithm.

### Use Exponential Models in Applications

In previous sections we were able to solve some applications that were modeled with exponential equations. Now that we have so many more options to solve these equations, we are able to solve more applications.
We will again use the Compound Interest Formulas and so we list them here for reference.
We have seen that growth and decay are modeled by exponential functions. For growth and decay we use the formula  Exponential growth has a positive rate of growth or growth constant, , and exponential decay has a negative rate of growth or decay constant, k.
We can now solve applications that give us enough information to determine the rate of growth. We can then use that rate of growth to predict other situations.
Radioactive substances decay or decompose according to the exponential decay formula. The amount of time it takes for the substance to decay to half of its original amount is called the half-life of the substance.
Similar to the previous example, we can use the given information to determine the constant of decay, and then use that constant to answer other questions.

### Key Concepts

1. One-to-One Property of Logarithmic Equations: For  and  is any real number:

2. Compound Interest:
For a principal, P, invested at an interest rate, r, for t years, the new balance, A, is:

3. Exponential Growth and Decay: For an original amount,  that grows or decays at a rate, r, for a certain time t, the final amount, A, is

### Section Exercises

### Practice Makes Perfect

Solve Logarithmic Equations Using the Properties of Logarithms
In the following exercises, solve for x.
Solve Exponential Equations Using Logarithms
In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.
In the following exercises, solve each equation.
In the following exercises, solve for x, giving an exact answer as well as an approximation to three decimal places.
Use Exponential Models in Applications
In the following exercises, solve.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

### Chapter Review Exercises

### Finding Composite and Inverse Functions

Find and Evaluate Composite Functions
In the following exercises, for each pair of functions, find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).
In the following exercises, evaluate the composition.
Determine Whether a Function is One-to-One
In the following exercises, for each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.
In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.
Find the Inverse of a Function
In the following exercise, find the inverse of the function. Determine the domain and range of the inverse function.
In the following exercise, graph the inverse of the one-to-one function shown.
In the following exercises, verify that the functions are inverse functions.
In the following exercises, find the inverse of each function.

### Evaluate and Graph Exponential Functions

Graph Exponential Functions
In the following exercises, graph each of the following functions.
Solve Exponential Equations
In the following exercises, solve each equation.
Use Exponential Models in Applications
In the following exercises, solve.

### Evaluate and Graph Logarithmic Functions

Convert Between Exponential and Logarithmic Form
In the following exercises, convert from exponential to logarithmic form.
In the following exercises, convert each logarithmic equation to exponential form.
Evaluate Logarithmic Functions
In the following exercises, solve for x.
In the following exercises, find the exact value of each logarithm without using a calculator.
Graph Logarithmic Functions
In the following exercises, graph each logarithmic function.
Solve Logarithmic Equations
In the following exercises, solve each logarithmic equation.
Use Logarithmic Models in Applications

### Use the Properties of Logarithms

Use the Properties of Logarithms
In the following exercises, use the properties of logarithms to evaluate.
In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.
In the following exercises, use the Power Property of Logarithms to expand each logarithm. Simplify, if possible.
In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.
Use the Change-of-Base Formula
In the following exercises, rounding to three decimal places, approximate each logarithm.

### Solve Exponential and Logarithmic Equations

Solve Logarithmic Equations Using the Properties of Logarithms
In the following exercises, solve for x.
Solve Exponential Equations Using Logarithms
In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.
Use Exponential Models in Applications

### Practice Test

In the following exercises, use properties of logarithms to write each expression as a sum of logarithms, simplifying if possible.
In the following exercises, use the Properties of Logarithms to condense the logarithm, simplifying if possible.
In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.


# Conics

## Introduction

Five, Four. Three. Two. One. Lift off. The rocket launches off the ground headed toward space. Unmanned spaceships, and spaceships in general, are designed by aerospace engineers. These engineers are investigating reusable rockets that return safely to Earth to be used again. Someday, rockets may carry passengers to the International Space Station and beyond. One essential math concept for aerospace engineers is that of conics. In this chapter, you will learn about conics, including circles, parabolas, ellipses, and hyperbolas. Then you will use what you learn to investigate systems of nonlinear equations.


# Conics

## Distance and Midpoint Formulas; Circles

In this chapter we will be looking at the conic sections, usually called the conics, and their properties. The conics are curves that result from a plane intersecting a double cone—two cones placed point-to-point. Each half of a double cone is called a nappe.
There are four conics—the circle, parabola, ellipse, and hyperbola. The next figure shows how the plane intersecting the double cone results in each curve.
Each of the curves has many applications that affect your daily life, from your cell phone to acoustics and navigation systems. In this section we will look at the properties of a circle.

### Use the Distance Formula

We have used the Pythagorean Theorem to find the lengths of the sides of a right triangle. Here we will use this theorem again to find distances on the rectangular coordinate system. By finding distance on the rectangular coordinate system, we can make a connection between the geometry of a conic and algebra—which opens up a world of opportunities for application.
Our first step is to develop a formula to find distances between points on the rectangular coordinate system. We will plot the points and create a right triangle much as we did when we found slope in Graphs and Functions. We then take it one step further and use the Pythagorean Theorem to find the length of the hypotenuse of the triangle—which is the distance between the points.
The method we used in the last example leads us to the formula to find the distance between the two points  and
When we found the length of the horizontal leg we subtracted  which is
When we found the length of the vertical leg we subtracted  which is
If the triangle had been in a different position, we may have subtracted  or  The expressions  and  vary only in the sign of the resulting number. To get the positive value-since distance is positive- we can use absolute value. So to generalize we will say  and
In the Pythagorean Theorem, we substitute the general expressions  and  rather than the numbers.

This is the Distance Formula we use to find the distance d between the two points  and

### Use the Midpoint Formula

It is often useful to be able to find the midpoint of a segment. For example, if you have the endpoints of the diameter of a circle, you may want to find the center of the circle which is the midpoint of the diameter. To find the midpoint of a line segment, we find the average of the x-coordinates and the average of the y-coordinates of the endpoints.
Both the Distance Formula and the Midpoint Formula depend on two points,  and  It is easy to confuse which formula requires addition and which subtraction of the coordinates. If we remember where the formulas come from, it may be easier to remember the formulas.

### Write the Equation of a Circle in Standard Form

As we mentioned, our goal is to connect the geometry of a conic with algebra. By using the coordinate plane, we are able to do this easily.
We define a circle as all points in a plane that are a fixed distance from a given point in the plane. The given point is called the center,  and the fixed distance is called the radius, r, of the circle.

This is the standard form of the equation of a circle with center,  and radius, r.
In the last example, the center was  Notice what happened to the equation. Whenever the center is  the standard form becomes
In the next example, the radius is not given. To calculate the radius, we use the Distance Formula with the two given points.

### Graph a Circle

Any equation of the form  is the standard form of the equation of a circle with center,  and radius, r. We can then graph the circle on a rectangular coordinate system.
Note that the standard form calls for subtraction from x and y. In the next example, the equation has  so we need to rewrite the addition as subtraction of a negative.
To find the center and radius, we must write the equation in standard form. In the next example, we must first get the coefficient of  to be one.
If we expand the equation from ,  the equation of the circle looks very different.

This form of the equation is called the general form of the equation of the circle.
If we are given an equation in general form, we can change it to standard form by completing the squares in both x and y. Then we can graph the circle using its center and radius.
In the next example, there is a y-term and a -term. But notice that there is no x-term, only an -term. We have seen this before and know that it means h is 0. We will need to complete the square for the y terms, but not for the x terms.

### Key Concepts

1. Distance Formula: The distance d between the two points  and  is

2. Midpoint Formula: The midpoint of the line segment whose endpoints are the two points  and  is

To find the midpoint of a line segment, we find the average of the x-coordinates and the average of the y-coordinates of the endpoints.
3. Circle: A circle is all points in a plane that are a fixed distance from a fixed point in the plane. The given point is called the center,  and the fixed distance is called the radius, r, of the circle.
4. Standard Form of the Equation a Circle: The standard form of the equation of a circle with center,  and radius, r, is

5. General Form of the Equation of a Circle: The general form of the equation of a circle is

### Practice Makes Perfect

Use the Distance Formula
In the following exercises, find the distance between the points. Write the answer in exact form and then find the decimal approximation, rounded to the nearest tenth if needed.
Use the Midpoint Formula
In the following exercises, ⓐ find the midpoint of the line segments whose endpoints are given and ⓑ plot the endpoints and the midpoint on a rectangular coordinate system.
Write the Equation of a Circle in Standard Form
In the following exercises, write the standard form of the equation of the circle with the given radius and center
In the following exercises, write the standard form of the equation of the circle with the given radius and center
For the following exercises, write the standard form of the equation of the circle with the given center with point on the circle.
Graph a Circle
In the following exercises, ⓐ find the center and radius, then ⓑ graph each circle.
In the following exercises, ⓐ identify the center and radius and ⓑ graph.

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


# Conics

## Parabolas

### Graph Vertical Parabolas

The next conic section we will look at is a parabola. We define a parabola as all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.
Previously, we learned to graph vertical parabolas from the general form or the standard form using properties. Those methods will also work here. We will summarize the properties here.
The graphs show what the parabolas look like when they open up or down. Their position in relation to the x- or y-axis is merely an example.
To graph a parabola from these forms, we used the following steps.
The next example reviews the method of graphing a parabola from the general form of its equation.
The next example reviews the method of graphing a parabola from the standard form of its equation,

### Graph Horizontal Parabolas

Our work so far has only dealt with parabolas that open up or down. We are now going to look at horizontal parabolas. These parabolas open either to the left or to the right. If we interchange the x and y in our previous equations for parabolas, we get the equations for the parabolas that open to the left or to the right.
The graphs show what the parabolas look like when they to the left or to the right. Their position in relation to the x- or y-axis is merely an example.
Looking at these parabolas, do their graphs represent a function? Since both graphs would fail the vertical line test, they do not represent a function.
To graph a parabola that opens to the left or to the right is basically the same as what we did for parabolas that open up or down, with the reversal of the x and y variables.
In the next example, the vertex is not the origin.
In , we see the relationship between the equation in standard form and the properties of the parabola. The How To box lists the steps for graphing a parabola in the standard form  We will use this procedure in the next example.
In the next example, we notice the a is negative and so the parabola opens to the left.
The next example requires that we first put the equation in standard form and then use the properties.

### Solve Applications with Parabolas

Many architectural designs incorporate parabolas. It is not uncommon for bridges to be constructed using parabolas as we will see in the next example.

### Key Concepts

1. Parabola: A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.

2. How to graph vertical parabolas

3. How to graph horizontal parabolas

### Practice Makes Perfect

Graph Vertical Parabolas
In the following exercises, graph each equation by using properties.
In the following exercises, ⓐ write the equation in standard form and ⓑ use properties of the standard form to graph the equation.
Graph Horizontal Parabolas
In the following exercises, graph each equation by using properties.
In the following exercises, ⓐ write the equation in standard form and ⓑ use properties of the standard form to graph the equation.
Mixed Practice
In the following exercises, match each graph to one of the following equations: ⓐ x2 + y2 = 64 ⓑ x2 + y2 = 49ⓒ (x + 5)2 + (y + 2)2 = 4 ⓓ (x − 2)2 + (y − 3)2 = 9 ⓔ y = −x2 + 8x − 15 ⓕ y = 6x2 + 2x − 1
Solve Applications with Parabolas

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


# Conics

## Ellipses

### Graph an Ellipse with Center at the Origin

The next conic section we will look at is an ellipse. We define an ellipse as all points in a plane where the sum of the distances from two fixed points is constant. Each of the given points is called a focus of the ellipse.
We can draw an ellipse by taking some fixed length of flexible string and attaching the ends to two thumbtacks. We use a pen to pull the string taut and rotate it around the two thumbtacks. The figure that results is an ellipse.
A line drawn through the foci intersect the ellipse in two points. Each point is called a vertex of the ellipse. The segment connecting the vertices is called the major axis. The midpoint of the segment is called the center of the ellipse. A segment perpendicular to the major axis that passes through the center and intersects the ellipse in two points is called the minor axis.
We mentioned earlier that our goal is to connect the geometry of a conic with algebra. Placing the ellipse on a rectangular coordinate system gives us that opportunity. In the figure, we placed the ellipse so the foci  are on the x-axis and the center is the origin.
The definition states the sum of the distance from the foci to a point  is constant. So  is a constant that we will call  so,  We will use the distance formula to lead us to an algebraic formula for an ellipse.
To graph the ellipse, it will be helpful to know the intercepts. We will find the x-intercepts and y-intercepts using the formula.
Notice that when the major axis is horizontal, the value of a will be greater than the value of b and when the major axis is vertical, the value of b will be greater than the value of a. We will use this information to graph an ellipse that is centered at the origin.
We summarize the steps for reference.
Sometimes our equation will first need to be put in standard form.

### Find the Equation of an Ellipse with Center at the Origin

If we are given the graph of an ellipse, we can find the equation of the ellipse.

### Graph an Ellipse with Center Not at the Origin

The ellipses we have looked at so far have all been centered at the origin. We will now look at ellipses whose center is
The equation is  and when  the major axis is horizontal so the distance from the center to the vertex is a. When  the major axis is vertical so the distance from the center to the vertex is b.
If we look at the equations of  and  we see that they are both ellipses with  and  So they will have the same size and shape. They are different in that they do not have the same center.

Notice in the graph above that we could have graphed  by translations. We moved the original ellipse to the right 3 units and then up 1 unit.
In the next example we will use the translation method to graph the ellipse.
When an equation has both an  and a  with different coefficients, we verify that it is an ellipsis by putting it in standard form. We will then be able to graph the equation.

### Solve Application with Ellipses

The orbits of the planets around the sun follow elliptical paths.

### Key Concepts

1. Ellipse: An ellipse is all points in a plane where the sum of the distances from two fixed points is constant. Each of the fixed points is called a focus of the ellipse.

If we draw a line through the foci intersects the ellipse in two points—each is called a vertex of the ellipse.
The segment connecting the vertices is called the major axis.
The midpoint of the segment is called the center of the ellipse.
A segment perpendicular to the major axis that passes through the center and intersects the ellipse in two points is called the minor axis.
2. Standard Form of the Equation an Ellipse with Center  The standard form of the equation of an ellipse with center  is

The x-intercepts are  and
The y-intercepts are  and
3. How to an Ellipse with Center

4. Standard Form of the Equation an Ellipse with Center  The standard form of the equation of an ellipse with center  is

When  the major axis is horizontal so the distance from the center to the vertex is a.
When  the major axis is vertical so the distance from the center to the vertex is b.

### Practice Makes Perfect

Graph an Ellipse with Center at the Origin
In the following exercises, graph each ellipse.
Find the Equation of an Ellipse with Center at the Origin
In the following exercises, find the equation of the ellipse shown in the graph.
Graph an Ellipse with Center Not at the Origin
In the following exercises, graph each ellipse.
In the following exercises, graph each equation by translation.
In the following exercises, ⓐ write the equation in standard form and ⓑ graph.
In the following exercises, graph the equation.
Solve Application with Ellipses

### Writing Exercises

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?