Sketching the Graph of Higher Degree Polynomial Functions

This video tutorial teaches how to sketch graphs of higher degree polynomials by finding zeros and determining end behavior. It covers the steps to find x-intercepts, explains the end behavior based on the degree and leading coefficient, and discusses sign changes at roots. The tutorial includes examples to illustrate these concepts, emphasizing the importance of understanding multiplicity and intervals for accurate graphing.

What is the first step in finding the x-intercepts of a polynomial function?

• Multiply the polynomial by a constant

• Integrate the polynomial

• Set the polynomial equal to zero

• Differentiate the polynomial

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

• Increases on both sides

• Decreases on both sides

• Decreases on the left and increases on the right

• Increases on the left and decreases on the right

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

• The sign stays the same

• The sign changes

• The function becomes undefined

• The function reaches a maximum

If a polynomial has a double root at x = 3, what happens to the sign of the function at this point?

• The function becomes zero

• The sign stays the same

• The function becomes infinite

• The sign changes

What is the first step in sketching a polynomial graph?

• Determine the end behavior

• Find the zeros and note their multiplicity

• Draw a rough sketch

• Calculate the derivative

How does a polynomial with a negative leading coefficient and an odd degree behave?

• Increases on the left and decreases on the right

• Increases on both sides

• Decreases on both sides

• Decreases on the left and increases on the right

What is the significance of the leading term in determining the end behavior of a polynomial?

• It changes the degree of the polynomial

• It dictates the behavior at the far left and right

• It affects the symmetry of the graph

• It determines the y-intercept