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
- 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?
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