The purpose of this lesson is to formally define what the expression for a polynomial function can look like when written in standard form and to expose students to a wide variety of graphs of polynomials. Students are introduced to the new terms degree, relative minimum, and relative maximum for describing features of expressions and graphs. The degree of a polynomial is the highest exponent occurring on \(x\) when the polynomial is written out as a sum of non-zero constants times powers of \(x\), with like terms collected. The relative minimum is a point on the graph of a function that is lower than any of the points around it, and the relative maximum is a point on the graph of a function that is higher than any of the points around it.
Students are also reminded of other useful vocabulary for polynomials, such as leading term and constant term, and use precise language to describe expressions and identify polynomials (MP6). In this lesson, students use vocabulary to make connections between polynomials and their graphs. They will continue to deepen their understanding of these connections throughout the unit.
In the second activity, students match equations to graphs by focusing on the structure (MP7). Students will continue this work in future lessons as they make connections between the structure of a polynomial written in factored form and the zeros of the function. In the final activity, students have the opportunity to play with graphing a variety of polynomials as they try to make graphs that meet specific criteria. This is meant to be informal and an opportunity for students to improve their skill graphing with technology while building their mental profile of what graphs of polynomials can look like.
- Identify relative minimums and relative maximums of graphs of polynomials.
- Identify the degree, leading term, and constant term of a polynomial.
- Let’s see what polynomials can look like.
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)
- I can identify important characteristics of polynomial graphs and expressions.
The degree of a polynomial in \(x\) is the highest exponent occuring on \(x\) when you write the polynomial out as a sum of non-zero constants times powers of \(x\) (with like terms collected).
A polynomial function of \(x\) is a function given by a sum of terms, each of which is a constant times a whole number power of \(x\). The word polynomial is used to refer both to the function and to the expression defining it.
A point on the graph of a function that is higher than any of the points around it.
A point on the graph of a function that is lower than any of the points around it.