Lesson 11

Finding Intersections

11.1: Math Talk: When $f$ Meets $g$ (5 minutes)


In this activity, students have an opportunity to notice and make use of structure (MP7) in order to identify a point where the graphs of two given functions intersect. The work here leads directly into the next activity in which students use algebraic methods to identify all points of intersection between two polynomials.


Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Mentally identify a point where the graphs of the two functions intersect, if one exists.

\(f(x)=x\) and \(g(x)=3\)

\(j(x)=(x+3)(x-3)\) and \(k(x)=0\)

\(m(x)=(x+3)(x-3)\) and \(n(x)=(x-3)\)

\(p(x)=(x+5)(x-5)\) and \(q(x)=(x+3)(x-3)\)

Student Response

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Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?”

If not brought up during the discussion, remind students that these are called systems of equations. Ask, “What are some ways you have solved systems of equations in the past?” (Graphing, substitution, elimination.) Tell students that in the next activity, they are going to think about how to use substitution to solve systems of equations in which at least one of the equations is a quadratic.

Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

11.2: More Points of Intersection (20 minutes)


The purpose of this lesson is for students to solve systems of equations involving quadratics. Unlike systems of linear equations, systems involving quadratic functions can have 0, 1, or 2 distinct solutions, in addition to the infinitely many solutions case. But as with linear equations, solutions can be identified from graphs as the \(x\)-values of the points where the lines cross. Students will think about solutions both algebraically and graphically in this lesson. Students begin the lesson by focusing on structure to identify possible solutions to a series of systems of equations (MP7). Building on this, students then combine their skills solving systems of equations using substitution and solving quadratic equations to identify all solutions to systems of equations involving at least one quadratic function. They will continue to use these skills in future lessons involving rational functions and polynomial identities.
The last activity is meant to make clear that while algebraic solving methods are useful, technology helps us identify solutions to systems of equations where the steps needed to solve by hand are less obvious. This activity also asks students to consider what a possible factor of a polynomial expression written in standard form could be, which will be the focus of the following lessons leading into the Remainder Theorem.


Arrange students in groups of 2. Display the equations \(a(x)=(x+2)(x-2)\) and \(b(x)=(x-2)\) for all to see, and ask students to try and solve the system without graphing. After quiet work time, have students share their work with their partner and reach agreement on the solutions. Invite 2–3 students to share their solution process before starting the activity.

After students have had 5 minutes to start working on the activity, ask them to pause, and display the following solution strategy for all to see, if it was not discussed earlier.

\(\displaystyle \begin {align*} (x+2)(x-2) &= (x-2) \\ (x+2)(x-2)-(x-2) &= 0 \\ (x-2)(x+2-1) &= 0 \\ (x-2)(x+1) &= 0 \\ \end {align*}\)

Ask, “What is happening at each step?” (Rewriting the equation so it is an expression equal to zero, then factoring the \((x-2)\) from each term.) Encourage students to look for structure as they are solving the remaining systems.

Since this activity was designed to be completed without technology, ask students to put away any devices.

Action and Expression: Internalize Executive Functions. To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Look for students who are correctly factoring, and distributing where applicable. If students are having trouble identifying an error, invite them to check if there are any steps they could rewrite to show the steps more fully. Encourage them to use parentheses to keep their expressions organized, and to show each step one at a time. If students are still having trouble identifying the error, color coding can be used to highlight structures and clarify steps.
Supports accessibility for: Memory; Organization

Student Facing

For each pair of polynomials given, find all points of intersection of their graphs.

  1. \(c(x)=x^2-7\) and \(d(x)=2\)
  2. \(f(x)=(x+7)(x-4)\) and \(g(x)=x-4\)
  3. \(m(x)=(x+7)(x-4)\) and \(n(x)=(2x+5)(x-4)\)
  4. \(p(x)=(x+1)(x-8)\) and \(q(x)=(x+2)(x-4)\)

Student Response

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Student Facing

Are you ready for more?

Find all points of intersection of the graphs of the equations \(p(x)=(2x+3)(x-5)\) and \(q(x)=(x+5)(x+1)(x-3)\). Use graphing technology to check your solutions.

Student Response

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Anticipated Misconceptions

Some students may divide by the same factor on each side of the equation during their solving process. For example, going from \((x+7)(x-4) = x-4\) in one step to \(x+7=1\) in the next. While this does result in one of the solutions, in this example, \(x=\text-6\), the other solution, \(x=4\), is lost. Encourage these students to try solving without dividing by terms with an \(x\) in them to see both solutions. It may be helpful to ask, “What would happen if \(x\) was 4? Would it then make sense to divide by \(x-4\)?” (No, because then we would be dividing each side by 0.)

Activity Synthesis

The purpose of this discussion is for students to understand the different ways their classmates solved the questions in the activity. Select 1 previously identified student per question to share their solutions. Encourage students to ask clarifying questions about why different solving steps were used. After each student shares, ask if any students solved the question a different way, and invite those students to share their steps.

Conclude the discussion by asking students if it is possible for a system of equations with two quadratic functions to have 3 solutions. If possible, display a graph with two quadratics whose shapes can be manipulated to help convince students that 2 is the maximum number of solutions for distinct quadratic functions.

Conversing: MLR8 Discussion Supports. Use this routine to help students reflect on the different ways their classmates solved the equations. For each strategy that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was able to accurately restate their thinking. Call students’ attention to any words or phrases that helped clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making

11.3: Graphing to Find Points of Intersection (10 minutes)


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 \(p(x)\).


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 \(q(x) = 5640\).

  1. Use graphing technology to find a value of \(x\) that makes \(p(x)=q(x)\) true.
  2. For the \(x\)-value at the point of intersection, what can you say about the value of \(5x^3+6x^2+4x-5640\)?
  3. What does your answer suggest is a possible factor of \(5x^3+6x^2+4x-5640\)?
    1. Write your own polynomial \(m(x)\) of degree 3 or higher.
    2. 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.

Lesson Synthesis

Lesson Synthesis

The purpose of this synthesis is for students to reflect on the similarities and differences when solving systems of equations in which both equations are linear and when solving systems of equations in which one equation is quadratic and the other is linear.

Display the following systems for all to see:

\(\displaystyle \begin {align*} y &=x+5 \\ y &= 5-x \\ \end {align*}\)

\(\displaystyle \begin {align*} y &=x+5 \\ y &= (x-2)(x+5) \\ \end {align*}\)

Ask students to solve both systems and then write a brief summary of how solving systems with quadratic equations is similar and different from solving systems with only linear equations. If time allows, invite students to share some of the things they wrote. Highlight any students that note the difference in the number of possible solutions and those who notice that the steps used to solve for \(x\) can be very similar.

11.4: Cool-down - Find Some More Points (5 minutes)


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Student Lesson Summary

Student Facing

When asked to find all values of \(x\) that make an equation like \((x+4)(x-8)=(2-x)(x-8)\) true, one way to consider the question is to ask where the graphs of the functions \(f(x)=(x+4)(x-8)\) and \(g(x)=(2-x)(x-8)\) intersect.

Graph of functions f of x and g of x.

Since the coordinate of any point of intersection has the form \((a,f(a))=(a,g(a))\), these points must make \(f(x)=g(x)\) true when \(x=a\). In our example, we can tell from the graph that both \(x=\text-1\) and \(x=8\) are solutions to the original equation.

We can also use algebra to identify solutions to \((x+4)(x-8)=(2-x)(x-8)\) by rearranging and then recognizing that both parts have a factor of \((x-8)\) in common:

\(\displaystyle \begin{align*} (x+4)(x-8)&=(2-x)(x-8)\\ (x+4)(x-8)-(2-x)(x-8)&=0\\ (x-8)(x+4-2+x)&=0\\ (x-8)(2x+2)&=0\\ x&=8, \text-1\\ \end{align*}\)

For polynomials created to model specific situations that have a more messy structure, solving without using technology can be challenging, especially because the graphs of two polynomials can intersect at multiple points because of the way they curve. Fortunately, this type of solving challenge is one that computer algebra systems are usually very good at, leaving the interpretation of the solution up to humans.