Graphing Rational Functions: Key Concepts and Techniques

The video tutorial covers graphing rational functions, which are functions of the form p(x)/q(x) where both p(x) and q(x) are polynomials. It outlines steps to simplify the function, find x-intercepts, vertical and horizontal asymptotes, and identify holes. The video provides rules for determining horizontal asymptotes based on the degrees of the numerator and denominator. It explains holes as removable discontinuities occurring when there is a common factor in the numerator and denominator. The tutorial includes several examples demonstrating these concepts in practice.

What is the first step in graphing a rational function?

• Identify any holes

• Simplify the function

• Find the vertical asymptote

• Find the horizontal asymptote

How do you find the x-intercept of a rational function?

• Find the leading coefficient

• Simplify the function

• Set the numerator equal to zero

• Set the denominator equal to zero

What determines the horizontal asymptote when the degree of the numerator is less than the degree of the denominator?

• The horizontal asymptote is at zero

• The leading coefficient of the denominator

• The leading coefficient of the numerator

• The ratio of the leading coefficients

What is a hole in the graph of a rational function?

• A horizontal asymptote

• A point where the function is undefined

• A vertical asymptote

• A removable discontinuity

How do you identify a hole in a rational function?

• Find a common factor in the numerator and denominator

• Set the numerator equal to zero

• Set the denominator equal to zero

• Find the leading coefficient

In the function f(x) = (x-4)/(x+4), what is the x-intercept?

• x = -1

• x = 0

• x = 4

• x = -4

For the function f(x) = (x-4)/(x+4), what is the vertical asymptote?

• x = 4

• x = -1

• x = -4

• x = 0

In the function f(x) = (2x-3)/(x-1), what is the horizontal asymptote?

• y = 1

• y = 0

• y = -1

• y = 2

What happens to the graph of a rational function if the only factor in the denominator cancels out?

• It becomes undefined

• It becomes a vertical asymptote

• It becomes a horizontal asymptote

• It becomes a linear equation with a hole

For the function f(x) = (x^2-1)/(x-1), what is the hole in the graph?

• x = 1

• x = 0

• x = 2

• x = -1