Characteristics of Rational Functions

The video tutorial explains how to write an equation for a rational function with specific characteristics, including vertical asymptotes at x = 5 and x = 3, x-intercepts at x = -2 and x = -6, and a horizontal asymptote at y = 8. The process involves setting up the function with binomial factors in the numerator and denominator, determining the value of 'a' based on the horizontal asymptote, and verifying the function's characteristics through graphing.

What are the vertical asymptotes of the rational function described in the video?

• x = 5 and x = 3

• x = -2 and x = -6

• y = 8

• x = 0

Where do x-intercepts occur in a rational function?

• Zeros of the numerator

• At the horizontal asymptote

• Zeros of the denominator

• Where the function is undefined

Which factors must be included in the denominator to achieve the given vertical asymptotes?

• (x + 2) and (x + 6)

• (x - 5) and (x - 3)

• (x - 2) and (x - 6)

• (x + 5) and (x - 3)

What is the role of 'a' in the rational function equation?

• It changes the degree of the function

• It sets the horizontal asymptote

• It adjusts the vertical asymptotes

• It determines the x-intercepts

What is the horizontal asymptote of the rational function?

• y = 0

• y = 1

• y = -8

• y = 8

What happens when the degree of the numerator and denominator are the same?

• The function is undefined

• The horizontal asymptote is determined by the leading coefficients

• The vertical asymptotes are at x = 0

• The function has no asymptotes

How do you verify the characteristics of a rational function graphically?

• By ensuring the graph passes through the origin

• By checking the x-intercepts

• By looking for symmetry

• By confirming the asymptotes and intercepts

What is the equation of the rational function described in the video?

• f(x) = (x - 2)(x - 6) / (x - 5)(x + 3)

• f(x) = 8(x - 2)(x - 6) / (x - 5)(x + 3)

• f(x) = (x + 2)(x + 6) / (x + 5)(x - 3)

• f(x) = 8(x + 2)(x + 6) / (x + 5)(x - 3)

Which of the following is NOT a characteristic of the rational function described?

• Horizontal asymptote at y = 8

• Vertical asymptotes at x = 5 and x = 3

• Vertical asymptote at x = -5

• X-intercepts at x = -2 and x = -6

Why is it important to verify the graph of a rational function?

• To check for errors in the equation

• To find additional intercepts

• To confirm it matches the given characteristics

• To ensure it is a straight line