Graphing Rational Functions: Key Concepts and Techniques

This video tutorial covers graphing rational functions, focusing on identifying and using asymptotes. It begins with rewriting rational functions to find asymptotes through long division. The tutorial then explains how to find vertical and horizontal asymptotes by examining the function's denominator and the highest power of x. It includes examples of graphing rational functions by identifying asymptotes and intercepts, and concludes with a complex example involving multiple asymptotes. The video emphasizes the role of asymptotes in determining the graph's shape and behavior.

What is the first step in rewriting the rational function g(x) = 4x / (x - 3) to identify asymptotes?

• Factor the numerator

• Complete the square

• Use synthetic division

• Use long division

In the function g(x) = 4 + 12 / (x - 3), what does the term '+4' represent?

• Shift down by 4

• Shift to the right by 4

• Shift up by 4

• Shift to the left by 4

How do you determine the vertical asymptotes of a rational function?

• Set the numerator equal to zero

• Set the denominator equal to zero

• Find the largest power of x in the numerator

• Find the largest power of x in the denominator

For the function with the denominator x^2 + 7x + 12, what are the vertical asymptotes?

• x = 3 and x = -4

• x = -3 and x = -4

• x = 3 and x = 4

• x = -3 and x = 4

What is the horizontal asymptote of the function 2x + 1 / (x - 5)?

• y = -5

• y = 1

• y = 0

• y = 2

What is the vertical asymptote of the function f(x) = 2x + 1 / (3x - 4)?

• x = -3/4

• x = 3/4

• x = -4/3

• x = 4/3

How do you find the x-intercept of the function f(x) = 2x + 1 / (3x - 4)?

• Find the largest power of x in the numerator

• Find the largest power of x in the denominator

• Set the numerator equal to zero

• Set the denominator equal to zero

What is the horizontal asymptote of the function f(x) = 2x + 1 / (3x - 4)?

• y = 0

• y = 3/2

• y = 2/3

• y = 1

For the function with the denominator factored as (x + 5)(x - 3), what are the vertical asymptotes?

• x = 5 and x = -3

• x = 5 and x = 3

• x = -5 and x = 3

• x = -5 and x = -3

What is the horizontal asymptote of the function with the numerator 4x^2 and the denominator x^2?

• y = 1

• y = 4

• y = 0

• y = -4