Math 89 Hybrid Course Notes Spring 2016 Instructor: Yolande Petersen How to use these notes The notes and example problems cover all content that I would normally cover in (f2f) course. If you can do every example problem, you should have reasonable coverage of the course. My personal recommendation: Look over the notes before each class. Do all the examples you know how to do easily, and make note of the ones that are difficult for you In class, take notes on the problems that are presented. If time permits, students may request to see a few additional difficult problems. After lecture, try to complete all the problems in the notes, using videos if necessary. If you still have trouble, post your question in the discussion forum, where a classmate or the instructor can answer it. Check your notes with the posted completed notes. Steps 1-5 are equivalent to what you would learn from previewing and attending class every day in a f2f course. Now you should be ready to complete the homework. Explanation of symbols: Asterisks (*) indicate problems that I intend to cover in class. Some problems without an asterisk will also be covered, if time permits. If time is tight, I may skip a * problem. When an example has 2 problems with “OR”, the first problem is what I would prefer to present in class, and the second is a similar problem with a Khan Academy video . If time allows, I will use the first example in class; otherwise the second example with video can be used to learn the concept. Problems with a corresponding Khan Academy video available are highlighted in blue. Problems with a corresponding Petersen video will be highlighted in yellow Problems with a corresponding Souza video will be highlighted in green R. 1 Introduction to Algebraic Expressions face
Topics include: Sets of numbers Natural #’s: {1, 2, 3, ….} Whole #’s: {0, 1, 2, 3, ….} Integers: {… - 3, - 2, - 1, 0, 1, 2, 3, ….} Rational #’s: {n| n = a/b, where a and b are integers, b 0} Order of Operations 1. Parentheses/Grouping Symbols (including radicals, absolute value, etc.) 2. Exponents 3. Multiply & Divide 4. Add & Subtract Example a: Simplify Laws/Properties: Commutative Addition: a + b = b + a Multiplication: ab = ba Associative Addition: (a + b) + c = a + (b + c) Multiplication: (ab)c = a(bc) Distributive: a(b + c) = ab + ac Identity Addition: 0 + a = a additive identity is ______ Multiplication: 1 x a = a multiplicative identity is ______ Inverse Addition: a + (-a) = 0 additive inverse is _____ Multiplication: a x ( 1/a) = 1 multiplicative inverse is _____ Multiplication by 0: 0 x a = Multiplication by -1: -1 x a = -1(a) = Equivalent fractions both reduced forms will generally be accepted on tests Combining like terms; Ex b Simplify: (4x2 + x – 5) – 2(x2 – 7x – 3) R. 2 Equations, Inequalities, and Problem-Solving Solving Equations - Isolate the Variable Use the Add. & Mult. Prop. of Equality to “undo” addition & multiplication Ex a Solve: 4(1 – 3x) = 9 – 7x
A2 – B2 = A3 + B3 = A3 – B3 = Cautions: A2 – B2 (A – B)2 A3 + B3 (A + B)3 A3 – B3 (A – B)3 Ex e Factor 40c3 – 5d3 Solving – set each factor = 0 Ex f s2 – 2s - 35 = 0 Ex g 4x2 = 20x - 25 R.6 Rational Expressions and Equations Rational Expression – , where Q(x)0 (polynomial fraction) Zeroes in the denominator are to be avoided. (Try 50 on your calculator) Zeroes in the denominator are sometimes called *Ex a Find the values where is undefined Reducing/Cancelling Common Factors Common factors (multiplied) can be cancelled Common terms (added or subtracted) cannot be cancelled
Procedure for Completing the Square Get equation in the form x2 + bx = c (x2 and x terms on left, number on right) Find the needed number to complete the square nn = Add it to both sides. Factor the perfect square. Write it as a square that looks like (x + b/2)2 Square root both sides Solve for x Ex f Solve: x2 – 8x – 2 = 0 Exg Solve: 4x2 – 40x - 300 = 0 (similar to Ex f, with Khan video) Ex h Solve 3k2 + k + 2 = 0 (This is nasty) 11.2 Quadratic Formula Recall: Quadratic equations have the standard form ax2 + bx + c = 0 Quadratic equations have 2, 1, or 0 real solutions Ex a Find a, b, and c for the equations: 3x2 – 2 = 5x x2 – 17 = 0
1 1 2 4 Ex a Graph the functions f(x), g(x), h(x) and p(x) above: Note: When “a” is positive, the parabola is ___________ When “a” is negative, the parabola is face ___________ All parabolas have either a “peak” called the _______________ or a “valley” called the _______________ Shifting - Graph f(x) = a(x – h)2 + k Ex b Graph f(x) = (x + 2)2 Ex c Graph y = -2(x – 2)2 + 5
2) Decomposition - Separating composite functions into smaller functions Assign one function to a “cluster” – where would you put parentheses? Ex c Express the following functions as the composition of functions. 1) h(x) = (2x + 3)3 2) h(x) = 3) h(x) =+ 4 (modified) It’s possible to decompose in more than one way Inverses and One-to-One Functions Function Never 2 outputs for same input Passes vertical line test – no vertical line cuts graph twice One-to-One – does not apply if it’s not a function Never has 2 inputs produce same output Passes horizontal line test – no horizontal line cuts graph twice. Which are functions? Which are one-to-one?: One-to-one functions have the properties Each f(x) has only one x associated with it, and each x has only one f(x) If f(x1) = f(x2), then x1 = x2 (if 2 outputs are the same, the inputs must be same) They are invertible (they have an inverse) , where f-1(x) is the inverse of f(x) Finding formulas for the inverse of a function We use the notation f-1(x) = “the inverse of f(x)” or “f-inverse of x” (NOT an exponent) Procedure for finding f-1(x) Let y = f(x) Exchange x and y (this creates the inverse relationship) Isolate y Replace y with f-1(x)
b) If 6% of the original amount of C-14 remains, how many years have passed? 13.1 Parabolas and Circles Quadratics often come in “stretched out” (general form) and “useful” (vertex) form Parabolas (vertical) Parabolas (horizontal) General form: y = ax2 + bx + c General: x = ay2 + by + c Vertex form: y = a(x – h)2 + k Vertex: x = a(y - k)2 + k or (y – k) = a(x – h)2 or (x – h) = a(y – k)2 Graphing Procedure (from Chapter 11) Decide direction of opening Find vertex Plot another point to find width Ex a Graph x = - y2 – 2y + 3 Which way does it face? Bonus material in videos (skip if you like): Vertex form can be written with the coefficient “a” = , so the vertex forms look like 4p(y – k) = (x – h)2 (vertical parabola) or 4p(x – h) = (y – k)2 (horizontal parabola) The value “p” is significant – it’s the distance from the vertex to the focus of the parabola. Ex 1-bonus: For the equation x2 + 8x = 4y – 8, write in standard form. Find the vertex, focus and graph (see video at https://www.youtube.com/watch?v=CKepZr52G6Y&ab_channel=Mathispower4u
Ex d xy = - 4 How would you rewrite this equation? Classifying Graphs Shape Standard Form ("useful" form) Landmarks General Form ("pretty" form) y = mx + b Ex: y = 3x + 5 Ax + By = C 3x – y = 5 (y – k) = a(x – h)2 Ex: 4(y – 3) = x2 Ax2 + Cx + Dy + E = 0 x2 – 4y + 12 = 0 (x – h) = a(y – k)2 Ex: 8(x – 3) = (y – 2)2 By2 + Cx + Dy + E = 0 y2 – 4y – 8x + 28 = 0 (x – h)2 + (y – k)2 = r2 Ex: (x+ 4)2 + (y – 6)2 = 25 x2 + y2 + Cx + Dy + E = 0 x2 + y2 + 8x – 12y + 43 = 0 Ex: Ax2 + By2 + Cx + Dy + E = 0 (coefficients of x2 & y2 same sign)
25x2 + 4y2 – 100x + 8y – 96 = 0 Ex: Ax2 - By2 + Cx + Dy + E = 0 (coefficients of x2 & y2 opposite signs) 4x2 – 9y2 – 18y – 45 = 0 Ex: By2 – Ax2 + Cx + Dy + E = 0 (coefficients of x2 & y2 opposite signs) x2 – y2 – 2x – 8y – 11 = 0 Identifying Quadratic (and Linear) Equations For each equation, tell what shape it is (line, circle, parabola, ellipse, hyperbola). x2 – 6x + 4y2 + 8y – 4 = 0 x2 – 6x – 4y2 + 8y – 4 = 0 x2 – 8x + y2 + 2y + 5 = 0 x – y = 9 x2 – y = 9 x2 – y2 = 9 x2 + y2 = 9 x2 + 9y2 = 9 4x2 + 8y2 = 32 x = 16y2 – 32