Graphing Rational Functions: Key Concepts and Techniques

This lesson covers graphing rational functions, including finding x-intercepts, vertical and horizontal asymptotes, holes, and slant asymptotes. It provides step-by-step examples to illustrate these concepts, emphasizing the importance of understanding polynomial degrees and leading coefficients in determining asymptotes.

What is the general form of a rational function?

• f(x) = P(x) / Q(x)

• f(x) = P(x) - Q(x)

• f(x) = P(x) * Q(x)

• f(x) = P(x) + Q(x)

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

• Set both numerator and denominator equal to zero and solve.

• Set the function equal to zero and solve.

• Set the numerator equal to zero and solve.

• Set the denominator equal to zero and solve.

How are vertical asymptotes determined?

• By setting the denominator equal to zero.

• By finding the degree of the numerator.

• By finding the degree of the denominator.

• By setting the numerator equal to zero.

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

• There is no horizontal asymptote.

• y = 1

• y = 0

• y equals the leading coefficient of the numerator divided by the leading coefficient of the denominator.

What happens when the degree of the numerator equals the degree of the denominator?

• The horizontal asymptote is y = 0.

• There is no horizontal asymptote.

• The horizontal asymptote is y equals the sum of the degrees.

• The horizontal asymptote is y equals the leading coefficient of the numerator divided by the leading coefficient of the denominator.

How do you find a hole in the graph of a rational function?

• Set a common factor from the numerator and denominator equal to zero and solve.

• Set the denominator equal to zero.

• Set the function equal to zero.

• Set the numerator equal to zero.

When does a slant asymptote occur?

• When the numerator and denominator have no common factors.

• When the degree of the numerator is less than the degree of the denominator.

• When the degree of the numerator equals the degree of the denominator.

• When the degree of the numerator is greater than the degree of the denominator.

What is the first step in graphing a rational function?

• Find the horizontal asymptotes.

• Find the x-intercepts.

• Find the vertical asymptotes.

• Factor the equation.

In the first example, what are the x-intercepts of the function x^2 - 9 / (x + 2)?

• x = -3 and x = 3

• x = -2 and x = 2

• x = -9 and x = 9

• x = -1 and x = 1

In the second example, what is the vertical asymptote of the function (x - 5) / (x - 1)?

• x = 5

• x = -1

• x = 1

• x = 0