Table of Contents

Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Lesson 9: Graphing polynomials

**Topic**. This lesson covers Session 9: Graphing polynomials

**Learning Outcomes.**

- Make connections between geometric features of polynomials (roots, extrema, end behavior) and corresponding algebraic features (factors, coefficients, etc).
- Translate between graphs, formulas, and natural language descriptions of polynomials.

**WeBWorK**. There is one WeBWorK assignment on today’s material:

- Polynomials – Graphs

**Additional Video Resources.**

**Question of the Day: **How can we find the formula for function by looking at the graph?

## Graphing Polynomials

In this lesson, we will explore the connections between the graphs of polynomial functions and their formulas. By the end of the lesson, you should be able to:

a) Look at the graph of a polynomial, estimate the roots and their multiplicities, identify extrema, and the degree of the polynomial, and make a guess at the formula.

b) Look at the formula of a polynomial and determine the roots and end behavior.

c) Write a formula for a polynomial meeting certain conditions (i.e. with certain roots, end behavior, etc.)

### Graphs of functions

Consider the graphs of the functions for different values of :

From the graphs, you can see that the overall shape of the function depends on whether is even or odd.

### Graphs of polynomials of degree 2

**Observation**. Let be a polynomial of degree . The graph of is a parabola.

- will have at most two roots.
- will have a single extrema (maximum or minimum, “turning point”)
- If ( is the number in front of the ) then opens upwards, if then opens downwards.

### Graphs of polynomials of degree 3

**Observation**. Let be a polynomial of degree 3.

- The graph may change its direction at most twice ( will have at most two extrema).
- will have at most 3 roots.
- If , then the graph goes up to the right, down to the left. If then the graph goes down to the right, up to the left.

### Graphs of polynomials of any degree

**Observation. **Let be a polynomial of degree .

- Then has at most roots, and at most extrema.

- If the degree of is even, : If the leading coefficient then the graph opens upwards, if the leading coefficient then the graph opens downwards.

- If the degree of is odd: If the leading coefficient , then the graph of goes up to the right, down to the left. If the leading coefficient , then the graph of goes down to the right, up to the left.

**Example.** Find a function of degree 3 with roots and where the root at has multiplicity two.

**Example.** Determine the polynomial function with leading coefficient of degree and having roots , and . Your final answer should not contain any ‘s.

*VIDEO: Finding polynomials to match polynomial descriptions*

*Video by Irania Vazquez*

#### Exit Question

Example. Without using a calculator, match each graph to its formula:

i)

ii)

iii)

#### Answer

Notice that graph a) goes up to the right and down to the left. From this we conclude that the function has *odd* degree, and also must have a *positive leading coefficient*. The only answer that fits these criteria is iii).

Graph b) must have *even degree* and a positive leading coefficient, so b) fits with i).

Graph c) has odd degree but must have a negative leading coefficient (since it goes down to the right and up to the left), which confirms that c) is ii).

Answer: a)-iii), b)-i), c)-ii)