Graphing Polynomial Functions: End Behavior and Zeros

The video tutorial explains how to graph polynomial functions by analyzing end behavior, multiplicity, and zeros. It covers even and odd functions, such as y = x^2, y = -x^2, y = x^3, and y = -x^3, and discusses how the sign of the leading coefficient affects the graph. The tutorial also explains the concept of multiplicity and its impact on graph behavior at x-intercepts, and provides an example problem to illustrate these concepts.

What is the end behavior of the graph of y = x^2?

• Up on the left, down on the right

• Down on the left, up on the right

• Up on both sides

• Down on both sides

How does the graph of y = -x^2 differ from y = x^2?

• It is shifted to the right

• It is reflected over the x-axis

• It is reflected over the y-axis

• It is shifted to the left

What is the end behavior of the graph of y = x^3?

• Up on both sides

• Up on the left, down on the right

• Down on the left, up on the right

• Down on both sides

For a polynomial with an odd degree and a negative leading coefficient, what is the end behavior?

• Down on both sides

• Down on the left, up on the right

• Up on the left, down on the right

• Up on both sides

What happens to the graph of a polynomial at an x-intercept with a multiplicity of 2?

• It crosses the x-axis in a straight line

• It bounces off the x-axis

• It has a horizontal tangent at the x-axis

• It does not touch the x-axis

If the multiplicity of an x-intercept is 3, how does the graph behave at that point?

• It does not touch the x-axis

• It bounces off the x-axis

• It crosses the x-axis in a straight line

• It has a horizontal tangent at the x-axis

What is the first step in graphing a polynomial function given in factored form?

• Find the zeros of the function

• Determine the end behavior

• Calculate the degree of the polynomial

• Identify the leading coefficient

How do you find the multiplicity of a zero in a polynomial function?

• By examining the exponent of the corresponding factor

• By looking at the leading coefficient

• By finding the degree of the polynomial

• By determining the end behavior

What is the overall degree of the polynomial y = (x + 2)(x - 1)^2(x - 4)?

• 2

• 5

• 3

• 4

In the example problem, why are the peaks of the graph symmetric?

• Because the zeros have different multiplicities

• Because the graph is symmetric about x = 1

• Because the leading coefficient is negative

• Because the polynomial is of odd degree