Graphing Polynomials: Zeros, Multiplicities, and End Behavior

This video tutorial teaches how to graph polynomials by analyzing zeros, end behavior, and multiplicity. It includes two examples: one with a polynomial in factored form and another with a more complex polynomial. The tutorial explains how to determine the end behavior using the leading coefficient and degree, and how multiplicity affects the graph's shape at intercepts. The video provides a step-by-step guide to sketching polynomial graphs and offers additional resources for further practice.

What are the key components to consider when graphing a polynomial?

• Intercepts, slope, and curvature

• Zeros, coefficients, and degree

• Zeros, end behavior, and multiplicity

• Roots, asymptotes, and limits

How do you find the zeros of a polynomial in factored form?

• By setting each factor equal to zero

• By finding the y-intercept

• By differentiating the polynomial

• By integrating the polynomial

What does a multiplicity of one indicate about a zero?

• The graph has a vertical asymptote

• The graph has a horizontal asymptote

• The graph crosses the x-axis

• The graph touches the x-axis and turns around

What does the leading coefficient of a polynomial tell us?

• The y-intercept of the graph

• The x-intercept of the graph

• The left end behavior of the graph

• The right end behavior of the graph

How does the degree of a polynomial affect its end behavior?

• Even degree means the graph has no end behavior

• Odd degree means same direction on both ends

• Even degree means opposite directions on both ends

• Odd degree means opposite directions on both ends

What happens to the graph at a zero with an even multiplicity?

• The graph crosses the x-axis

• The graph touches the x-axis and turns around

• The graph has a horizontal asymptote

• The graph has a vertical asymptote

How can you find the y-intercept of a polynomial?

• By finding the leading coefficient

• By setting y to zero and solving for x

• By finding the degree of the polynomial

• By setting x to zero and solving for y

In the second example, what are the x-intercepts of the polynomial f(x) = -3(x + 3)(x - 3)^3?

• 0 and -3

• 0 and 3

• 3 and -3

• -3 and 3

What is the degree of the polynomial f(x) = -3(x + 3)(x - 3)^3?

• 5

• 3

• 4

• 2

What does a multiplicity of three indicate about a zero?

• The graph has a horizontal asymptote

• The graph has a vertical asymptote

• The graph crosses the x-axis and bends

• The graph touches the x-axis and turns around