Section 7.2: Graph Simple Rational Functions

Front

What is a vertical asymptote?

Back

A vertical asymptote is a line x = a where a function approaches infinity or negative infinity as the input approaches a. It occurs when the denominator of a rational function equals zero.

Front

How do you find vertical asymptotes in a rational function?

Back

To find vertical asymptotes, set the denominator of the rational function equal to zero and solve for x.

Front

What is a rational function?

Back

A rational function is a function that can be expressed as the ratio of two polynomials, typically in the form f(x) = P(x)/Q(x), where P and Q are polynomials.

Front

What is the horizontal asymptote of a rational function?

Back

The horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches infinity or negative infinity. It is determined by the degrees of the numerator and denominator.

Front

What is the significance of the degrees of the numerator and denominator in determining horizontal asymptotes?

Back

If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degrees are equal, the horizontal asymptote is y = leading coefficient of numerator / leading coefficient of denominator.

Front

What is the relationship between the vertical asymptotes and the factors of the denominator?

Back

Vertical asymptotes correspond to the values of x that make the denominator zero, which are the roots of the denominator's factors.

Front

What is the difference between vertical and horizontal asymptotes?

Back

Vertical asymptotes are vertical lines where the function approaches infinity, while horizontal asymptotes are horizontal lines that the function approaches as x goes to infinity.

Front

How can you identify the vertical asymptotes from the function f(x) = (x+5)/x?

Back

For the function f(x) = (x+5)/x, the vertical asymptote is found by setting the denominator x = 0, which gives the vertical asymptote at x = 0.

Front

What are the steps to graph a rational function?

Back

1. Identify vertical and horizontal asymptotes. 2. Determine intercepts (x and y). 3. Analyze the behavior of the function around the asymptotes. 4. Plot additional points if necessary and sketch the graph.

Front

What is the y-intercept of a rational function?

Back

The y-intercept of a rational function is found by evaluating the function at x = 0, provided the function is defined at that point.