standard_description: Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^(2) + y^(2))^(2) = (x^(2 )– y^(2))^(2) + (2xy)^(2 )can be used to generate Pythagorean triples.
problem_activity: 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.
[IMG]
She says, “This graph looks like a parabola, so it must be a quadratic.”
Is Mai correct? Use graphing technology to check.
Explain how you could select a viewing window before graphing an expression like
\(p(x)\)
that would show the main features of a graph.
Using your explanation, what viewing window would you choose for graphing
\(f(x)=(x+1)(x-1)(x-2)(x-28)\)
?
Student Response
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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
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for free access to Extension Student Response.
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.


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