options: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
problem_activity: Activity
This activity revisits a polynomial students first encountered in an earlier lesson in this unit, relating the polynomials to the integers. The purpose of this activity is to make clear that technology allows us to identify solutions to equations that can be tedious to do by hand. The last question reinforces this point by asking students to write their own polynomial and find the
\(x\)
value that corresponds to a certain output, since they will need to estimate the solution from a graph.
The question about possible factors is meant to preview the work of the next several lessons in which students will identify factors of polynomial expressions written in standard form and eventually show that for a polynomial
\(p(x)\)
, if
\(p(a)=0\)
, then
\((x-a)\)
is a factor of
\(p(x)\)
.
Launch
Display functions
\(p\)
and
\(q\)
for all to see. Ask, “Without using a graphing calculator, what is the most points of intersection the two functions could have? The least?” Encourage students to make sketches to justify that 3 is the most and 1 is the least.
Provide access to devices that can run Desmos or other graphing technology.
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to spark students’ curiosity about a polynomial written in standard form. Display only the first sentence of this problem (“Consider the functions . . . .”) and ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving visualizing the graph, points of intersection, or possible factors. This will help students create the language of mathematical questions before feeling pressure to produce solutions.
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, to help students see how questions 1 and 2 are related, ask them to find the solutions for an equation such as
\(x^2-4=12\)
. Then, invite them to briefly reflect on the steps used. Next, ask them what they would do differently if they were told to find the
\(x\)
-values where two functions
\(f(x)= x^2-4\)
and
\(g(x)=12\)
intersect. Ask students to notice what elements were similar in each process.
Supports accessibility for: Social-emotional skills; Conceptual processing
Student Facing
Consider the functions
\(p(x) = 5x^3+6x^2+4x\)
and
\(q(x) = 5640\)
.
Use graphing technology to find a value of
\(x\)
that makes
\(p(x)=q(x)\)
true.
For the
\(x\)
-value at the point of intersection, what can you say about the value of
\(5x^3+6x^2+4x-5640\)
?
What does your answer suggest is a possible factor of
\(5x^3+6x^2+4x-5640\)
?
Write your own polynomial
\(m(x)\)
of degree 3 or higher.
Use graphing technology to estimate the values of
\(x\)
that make
\(m(x)= q(x)\)
true.
Student Response
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Anticipated Misconceptions
If students have trouble using the value of
\(5x^3+6x^2+4x-5640\)
at the
\(x\)
-value where the functions intersect to find a possible factor of
\(5x^3+6x^2+4x-5640\)
, encourage them to think about the relationship between factors of an expression and
\(x\)
-values for which the value of the expression is 0.
Activity Synthesis
The goal of this discussion is for students to see how to use technology to answer an algebraic question involving a higher degree polynomial. Invite students to share the polynomial
\(m(x)\)
that they wrote. If possible, display several versions of
\(m(x)\)
for all to see, and use the graph to estimate the
\(x\)
-values where
\(m(x)=q(x)\)
. If this is not possible, ask students to share the solutions they estimated. Students should understand that not all polynomials can be factored by hand, and that technology is a valuable tool for helping us interpret such polynomials.
Then ask students to suggest possible factors of
\(5x^3+6x^2+4x-5640\)
. After a few suggestions, ask students how they could prove something is a factor. It is important that students understand that for now, the only way to prove something like
\((x-10)\)
is a factor of
\(5x^3+6x^2+4x-5640\)
is to find the quadratic function that satisfies
\(5x^3+6x^2+4x-5640=(x-10)(ax^2+bx+c)\)
, where
\(a\)
,
\(b\)
, and
\(c\)
are real numbers. Tell students that the next lessons are about how to do this.


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