Graphing polynomials
• Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior (A-APR.B.3, F-IF.C.7c$^\star$).
• Use the remainder theorem to find factors of polynomials (A-APR.B.2, A-APR.D.6).
The last activity in the previous section prepared students to start viewing polynomials in one variable as defining functions. In this section they study the graphs of polynomial functions. They see that the long run behavior of a polynomial is determined by its highest degree term. The use the relationship between factors and zeros to sketch the graph of a polynomial or to choose an appropriate viewing window for a graph produced by technology. The learn The Remainder Theorem and use it to find factors of polynomials.
Tasks
1
Graphs of Power Functions
WHAT: This task asks students to graph the functions $y=x^2$, $y=x^3$, $y=x^4$ and $y=x^5$ and make comparisons between the graphs. Students analyze differences and similarities between functions with even and odd powers. They also look at points of intersection and relate these points to the equations.
WHY: This task gives a nice starting point to think about how the degree of a polynomial function can affect its shape and general behavior. The task does not direct students to use technology, but they could easily opt to carry out the graphing and marking points with technology (MP.5).
2
Running Time
WHAT: Students are given two polynomials, one cubic and one quadratic, that model the running time of two algorithms as a function of the input. They evaluate the polynomials at a given point, then graph and compare them. They decide which algorithm will have a better running time for various sized images.
WHY: This task highlights the idea that a polynomial’s long-term behavior is controlled by its leading term. When comparing the two polynomials, students see that although the cubic polynomial may have a shorter running time for small values of the input, the quadratic will always have a shorter running time once the input passes a certain threshold. In order to focus on different aspects of the situation, students must be flexible in adjusting their graphing window (MP.5).
3
Graphing from Factors I
WHAT: Students graph two polynomials with the same factors but different leading coefficient, and observe that they have the same x-intercepts. They explain this observation in terms of the structure of factored form, and then produce a polynomial with given x-intercepts.
WHY: The purpose of this task is to begin a progression of ideas linking the factored form of a polynomial $p(x)$ with its zeroes and with the x-intercepts of its graph. In this task students are lead to explain that if $p(x)$ has a factor $x – a$ then $p(a)$ is zero times some other number, and therefore must be zero. This means that the point $(a, 0)$ is on the graph of $y = p(x)$, so the graph cross the $x$-axis at $x = a.$
4
Graphing from Factors II
WHAT: Students are given a situation where a student has graphed a cubic polynomial in factored form using a viewing window that makes it appear to be a parabola. They explain how the factored form can be used to find a better viewing window that shows all of the x-intercepts.
WHY: In the previous task students were given the viewing window and asked to observe the connection between the intercepts and the factored form. In this case they see that they can use the factored form to anticipate the location of the intercepts and thus choose an appropriate viewing window. This further develops their understanding of the connection between factors and zeros.
5
Graphing from Factors III
WHAT: Students are given a cubic polynomial for which one of the roots is easy to observe because the coefficients add to 0, meaning that 1 must be a zero of the function. They deduce that $x – 1$ must be a factor, and then completely factor the polynomial. They use the factored form to sketch a graph (MP.7).
WHY: The purpose of this task is to introduce students to the Remainder Theorem and its use in factoring a polynomial. Students have previous experience recognizing that a linear factor of a polynomial gives rise to a root of the polynomial: if $x- a$ is a factor of $p(x)$ then $p(a) = 0$. Now they use this connection in reverse; if $p(a) = 0$ then $x - a$ is a factor. Students may not recognize that this reverse statement is not in fact equivalent to the original, or that it needs proving (it is the essence of the Remainder Theorem). It is worth pointing that out, but for now they are simply using the fact. The task also gives students experience with dividing a linear factor into a polynomial to find the remaining quadratic factor, using either long division or by guessing and checking (which can be a good preparation for the long division algorithm).
6
Zeroes and factorization of a quadratic polynomial I
WHAT: Students work through two special cases of the remainder theorem for a quadratic polynomial $p(x)$ where it can be seen directly. First they see that if $p(0) = 0$ then the constant term of the polynomial must be 0, so that $x$ (that is, $x- 0$) is a factor of $p(x)$. Then they use long division to see that if $p(1) = 0$ then $x- 1$ must be a factor. Then they are guided through the reasoning for the general case.
WHY: The purpose of this task is to give students a gentle introduction to the proof of the remainder theorem. Students who are prepared to see a proof of the theorem in general can work on the more advanced task "A-APR Zeros and factorization of a general polynomial" (https://www.illustrativemathematics.org/content-standards/tasks/788).