Lesson 7

Using Factors and Zeros

7.1: More Than Factors (5 minutes)

Warm-up

The purpose of this activity is to revisit connections made in the previous lesson about which parts of an expression do, and which do not, affect the intercepts of a graph of the function while preparing students for the activities that follow, in which they will consider appropriate graphing windows for specific polynomial functions.

Student Facing

\(M\) and \(K\) are both polynomial functions of \(x\) where \(M(x)=(x+3)(2x-5)\) and \(K(x)= 3(x+3)(2x-5)\).

  1. How are the two functions alike? How are they different?
  2. If a graphing window of \(\text-5 \leq x \leq 5\) and \(\text-20 \leq y \leq 20\) shows all intercepts of a graph of \(y=M(x)\), what graphing window would show all intercepts of \(y=K(x)\)?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Select students to share their ideas for how the two functions are alike and different. When discussing the last question, it is important students understand that there are many windows that work to see all the intercepts. Since the extra 3 in the second equation changes the vertical position of the points of the graph, however, only the vertical window size needs adjusting. Sometimes when adjusting windows, students change more than they need to and end up with a window that is either too zoomed-in or too zoomed-out on the vertical or horizontal scale (or both) to see important features of the graph.

7.2: Choosing Windows (15 minutes)

Activity

The goal of this activity is for students to reason about what horizontal intercepts a graph of a polynomial must have based on the structure of the factored form of the polynomial (MP7). Students begin by considering a polynomial equation and graph and reasoning about why the graph window is insufficient since it only shows two out of three intercepts. Students then consider the horizontal intercepts of a different polynomial through the lens of selecting an appropriate viewing window before checking with graphing technology.

Launch

Provide access to devices that can run Desmos or other graphing technology. 

Reading, Conversing, Writing: MLR5 Co-Craft Questions. To help students consider the graph window of a polynomial, start by only displaying Mai’s graph and comment. Give students 1–2 minutes to write their own mathematical questions about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving the terms “parabola” or “quadratic,” along with any questions about the scale on the axes. This will help students create questions about Mai’s statement and graph as they explore an appropriate graph window.
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Engagement: Develop Effort and Persistence. Break the class into small discussion groups and then invite a representative from each group to report back to the whole class. Ask that each discussion group member is equally prepared to report back to the whole group. As students work, encourage groups to use peer supports to ensure all group members are prepared to report out if selected.
Supports accessibility for: Language; Social-emotional skills; Attention

Student Facing

Mai graphs the function \(p\) given by \(p(x)=(x+1)(x-2)(x+15)\) and sees this graph.

Graph of polynomial function, xy-plane, origin O.

She says, “This graph looks like a parabola, so it must be a quadratic.”

  1. Is Mai correct? Use graphing technology to check.
  2. Explain how you could select a viewing window before graphing an expression like \(p(x)\) that would show the main features of a graph.
  3. Using your explanation, what viewing window would you choose for graphing \(f(x)=(x+1)(x-1)(x-2)(x-28)\)?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Select some different windows for graphing the function \(q(x) = 23(x-53)(x-18)(x+111)\). What is challenging about graphing this function?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students have trouble deciding what the constant term of \(f(x)\) is, remind them that it can be found quickly by multiplying the constant terms of each linear factor. Students can use the distributive property to understand why this is true.

Activity Synthesis

Select students to share how they selected a viewing window for \(f(x)=(x+1)(x-1)(x-2)(x-28)\). In particular, ask students to state what structure in the factored form they focused on in order to select their window size.

If students question whether we can really be sure we’re seeing all the horizontal intercepts, tell them that in future lessons, they will learn more about what functions are doing to the far left and right of a graph and how we can be sure there are no other horizontal intercepts outside our viewing window. Later in this unit, they will see that all zeros of a function correspond to factors of the function.

7.3: What’s the Equation? (15 minutes)

Activity

This activity asks students to write possible equations for polynomials with specific horizontal intercepts, using their understanding that a factor of \((x-a)\) means that \(a\) is a zero of the function and therefore \((a,0)\) is a horizontal intercept. This is the first time students are asked to write equations this way, and the use of graphing technology to quickly check that the graphs of their equations have the desired features is encouraged.

Monitor for students including constant multipliers or using factors of the form \(ax-b\) where \(\frac{b}{a}\) is the horizontal intercept to share their equations during the discussion.

Launch

Provide access to devices that can run Desmos or other graphing technology.

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 translating coordinates into factors. If students are stuck, invite them to identify the \(x\)-coordinate by completing the mathematical statement “\(x = \) _____ .” Then, ask students how they could manipulate this equation to find a possible factor (rearrange it so that one side is zero).
Supports accessibility for: Memory; Organization

Student Facing

Write a possible equation for a polynomial whose graph has the following horizontal intercepts. Check your equation using graphing technology.

  1. \((4, 0)\)
  2. \((0, 0)\) and \((4, 0)\)
  3. \((\text-2, 0)\), \((0,0)\) and \((4,0)\)
  4. \((\text-4,0), (0,0)\), and \((2,0)\)
  5. \((\text-5, 0)\), \(\left(\frac12, 0 \right)\), and \((3,0)\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may think that, for example, a zero at \(x=2\) means that \((x+2)\) is a possible factor. Encourage these students to use graphing technology to check their work.

Activity Synthesis

Display for all to see the five sets of horizontal intercepts from the activity. Throughout the discussion, add on equations that meet the criteria to help make clear that there are many equations that work. Select previously identified students to share their equations and reasoning about why their equations have the required horizontal intercepts. If not brought up by a student, ask about different factors that lead to a horizontal intercept of \(\frac12\). Make sure students understand that while both \((x-\frac12)\) and \((2x-1)\) are equal to zero when \(x=\frac12\), they are not identical and lead to different equations.

Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole-class discussion. Ask students to prepare a visual display that shows their mathematical thinking and reasoning for the polynomial with horizontal intercepts \((\text-2, 0)\), \((0, 0)\) and \((4,0)\). The display should include the equation, graph, and explanation. Invite students to quietly circulate and read at least 2 other posters or visual displays in the room. Give students quiet think time to consider what is the same and what is different about the displays. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify observations for choosing a specific function with the desired zeros. This will help students understand that there are many different equations that produce a graph with the same horizontal intercepts.
Design Principle(s): Cultivate conversation

Lesson Synthesis

Lesson Synthesis

Ask students to consider what they have learned so far about the relationship between the features of graphs and equations of polynomial functions. After a brief quiet think time, record student responses for all to see. Encourage students to use precise language when sharing. Possible responses:

  • If \(x-2\) is a factor of a polynomial, \(x=2\) is a zero of the polynomial.
  • Polynomials can have the same zeros and degree but still have different equations.
  • You can write equations of polynomials in different ways that help you see different features more easily, like the constant term which tells you the vertical intercept on the graph of the polynomial.

7.4: Cool-down - A Possible Polynomial Equation (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

We can use the zeros of a polynomial function to figure out what an expression for the polynomial might be.

Let’s say we want a polynomial function \(Z\) that satisfies \(Z(x)=0\) when \(x\) is -1, 2, or 4. We know that one way to write a polynomial expression is as a product of linear factors. We could write a possible expression for \(Z(x)\) by multiplying together a factor that is zero when \(x=\text-1\), a factor that is zero when \(x=2\), and a factor that is zero when \(x=4\). Can you think of what these three factors could be?

It turns out that there are many possible expressions for \(Z(x)\). Using linear factors, one possibility is \(Z(x)=(x+1)(x-2)(x-4)\). Another possibility is \(Z(x)=2(x+1)(x-2)(x-4)\), since the 2 (or any other rational number) does not change what values of \(x\) make the function equal to zero.

To check that these expressions match what we know about \(Z\), we can test the three values -1, 2, and 4 to make sure that \(Z(x)\) is 0 for those values. Alternatively, we can graph both possible versions of \(Z\) and see that the graphs intercept the horizontal axis at -1, 2, and 4, as shown here.

Graph of 2 polynomial functions, xy-plane, origin O.