Polynomial Graph Behavior and Analysis

This lesson teaches how to determine polynomial equations by analyzing graphs. It covers end behavior based on degree and lead coefficient, examines roots and intervals, and provides steps for graphing higher degree polynomials. The lesson also explores deriving possible equations from given graphs.

What is the first step in determining the equation of a polynomial from its graph?

• Find the degree of the polynomial

• Analyze the end behavior

• Identify the y-intercept

• Determine the lead coefficient

How does a polynomial with an even degree and a positive lead coefficient behave?

• Increases on the left and decreases on the right

• Decreases on both sides

• Increases on both sides

• Decreases on the left and increases on the right

What happens to the sign of a polynomial at an odd root?

• The polynomial becomes undefined

• The polynomial becomes zero

• The sign changes

• The sign remains the same

If a polynomial has a zero at x = 3 with a double root, what does this imply about the graph at x = 3?

• The graph passes through the x-axis

• The graph bounces off the x-axis

• The graph is undefined at x = 3

• The graph has a vertical asymptote at x = 3

What is the first step in graphing a higher degree polynomial?

• Determine the y-intercept

• Sketch the graph

• Find the zeros and note their multiplicities

• Analyze the end behavior

When sketching a polynomial graph, what should be done after finding the zeros?

• Determine the y-intercept

• Calculate the derivative

• Connect the points with a smooth curve

• Analyze the end behavior

How can you identify a polynomial's end behavior from its graph?

• By checking the symmetry

• By counting the number of roots

• By analyzing the lead term

• By looking at the y-intercept

What is a crucial factor to consider when finding a polynomial equation from a graph?

• The color of the graph

• The number of turning points

• The zeros and their multiplicities

• The slope of the tangent at the origin

If a graph behaves like negative x squared, what does this indicate about the lead coefficient?

• It is positive

• It is zero

• It is negative

• It is undefined

What adjustment is needed if the calculated y-intercept does not match the graph's y-intercept?

• Change the degree of the polynomial

• Add a constant to the function

• Adjust the lead coefficient

• Multiply the entire function by a constant