Graphing Rational Functions: Key Concepts and Techniques

This video tutorial provides a comprehensive guide to graphing rational functions. It covers three examples, each illustrating different aspects of graphing such as factoring, identifying holes, and determining vertical, horizontal, and slant asymptotes. The video also explains how to find x and y intercepts and plot additional points for a complete graph. The tutorial is designed to help learners understand the process of graphing rational functions step by step.

What is the first step in graphing a rational function?

• Identify the horizontal asymptote

• Factor the numerator and denominator

• Find the x-intercept

• Plot additional points

How do you determine if there are any holes in the graph?

• Set the denominator equal to zero

• Set the numerator equal to zero

• Look for factors in the numerator that cancel with factors in the denominator

• Check if the degrees of the numerator and denominator are equal

What is the vertical asymptote for the function y = (x-1)/(x^2-2x-3)?

• x = 1

• x = -1

• x = -3

• x = 3

How do you find the horizontal asymptote of a rational function?

• Find the x-intercept

• Set the denominator equal to zero

• Compare the degrees of the numerator and denominator

• Set the numerator equal to zero

What is the y-intercept of the function y = (x-1)/(x^2-2x-3)?

• 1/3

• 0

• 1

• -1

What happens to the graph near a vertical asymptote?

• It intersects the x-axis

• It crosses the asymptote

• It approaches positive or negative infinity

• It becomes undefined

In the second example, what is the location of the hole in the graph?

• (-4, 8/9)

• (4, -8/9)

• (4, 8/9)

• (-4, -8/9)

What is the vertical asymptote for the function y = (x^2-16)/(2x^2+7x-4)?

• x = 1/2

• x = 4

• x = -1/2

• x = -4

How do you find the slant asymptote of a rational function?

• By performing polynomial long division

• By finding the x-intercept

• By setting the denominator to zero

• By factoring the numerator

What is the slant asymptote for the function y = (x^3+x^2-2x)/(x^2+2x-8)?

• y = x + 1

• y = x - 1

• y = x - 2

• y = x + 2