Graphing Polynomial Functions

Overview

This lecture covers graphing polynomial functions by analyzing end behavior, multiplicity, and zeros, with step-by-step examples.

The end behavior of a polynomial depends on its degree (even or odd) and the sign of its leading coefficient.
For even degree and positive leading coefficient, both ends go up (up, up).
For even degree and negative leading coefficient, both ends go down (down, down).
For odd degree and positive leading coefficient, left end goes down and right end goes up (down, up).
For odd degree and negative leading coefficient, left end goes up and right end goes down (up, down).
End behavior can be described as x → ±∞ and corresponding y behavior.

Multiplicity is the exponent of a zero's factor in the polynomial.
Multiplicity 1: graph crosses the x-axis in a straight (linear) manner at the zero.
Multiplicity 2: graph touches (bounces off) the x-axis at the zero and does not cross.
Multiplicity 3: graph crosses the x-axis with a horizontal (flattened) tangent at the zero.

Set the polynomial (in factored form) equal to zero and solve each factor for x to find the zeros.
The exponent of each factor indicates the zero’s multiplicity.

Zeros: x = –2 (multiplicity 1), x = 1 (multiplicity 2), x = 4 (multiplicity 1).
Overall degree: add exponents (1 + 2 + 1 = 4), so this is a degree 4 polynomial (even degree).
Positive leading coefficient indicates end behavior is up on both sides (up, up).
At x = –2 and x = 4 (multiplicity 1), the graph crosses the x-axis in a straight line.
At x = 1 (multiplicity 2), the graph bounces off the x-axis.
Relative peak heights between zeros can be found by substituting x-values into the polynomial.

End Behavior — How the y-values of a polynomial behave as x approaches ±∞.
Multiplicity — The exponent of a zero's factor; determines if the graph crosses or bounces at the zero.
Zero (x-intercept) — Value of x where the polynomial equals zero (the graph touches or crosses the x-axis).
Leading Coefficient — The coefficient of the highest-degree term in a polynomial.
Degree — The highest exponent in a polynomial; determines the general shape and end behavior.

Practice describing end behavior and identifying multiplicity in given polynomials.
Work on additional graphing problems, finding zeros and their multiplicities.
Plug in values between zeros to check the graph’s relative peak heights.