Rational Functions Worksheet with Answers – Algebra 2 (Free PDF)

Problem 1 Finding the domain ★★★

State the domain of each function.

1. f(x) = 5x − 3
2. g(x) = x + 1x² − 16
3. h(x) = 72x + 5

Problem 2 Asymptotes from the equation ★★★

Give the equations of the vertical and horizontal asymptotes.

1. y = 2x + 4 − 3
2. y = –1x − 5 + 6

Problem 3 Describing transformations ★★★

Describe how the graph of y = 1x + 3 − 5 is obtained from the graph of y = 1x.

Problem 4 Reading a graph ★★★

The graph of a rational function is shown. State the equations of its asymptotes, its domain, and its range.

Problem 5 Inverse variation constant ★★★

The quantity y varies inversely with x, and y = 12 when x = 3. Write the equation and find y when x = 9.

Problem 6 A hole in the graph ★★★

Simplify f(x) = x² − 36x − 6 and find the coordinates of the hole.

Problem 7 A hole and an asymptote ★★★

For f(x) = x + 2(x + 2)(x − 5), find the hole and the vertical asymptote.

Problem 8 Horizontal asymptotes by degree ★★★

Find the horizontal asymptote of each function, or state that there is none.

1. f(x) = 3x² + 1x² − 4
2. g(x) = 5x + 2x² + 1
3. h(x) = x³2x² + 1

Problem 9 Writing an equation from features ★★★

A rational function y = ax − h + k has vertical asymptote x = −3, horizontal asymptote y = 2, and passes through (0, 3). Find a and write the equation.

Problem 10 Intercepts and asymptotes ★★★

For y = 2x − 6x + 1, find the x-intercept, the y-intercept, and both asymptotes.

Problem 11 Sketching a rational function ★★★

Sketch the graph of y = 2x + 1 − 3. Draw both asymptotes and plot at least four points.

Problem 12 Is it inverse variation? ★★★

The table shows pairs from a situation. Show that y varies inversely with x, write the equation, and find y when x = 15.

Problem 13 Rewriting in standard form ★★★

Divide to rewrite y = 3x + 5x + 2 in the form y = ax − h + k, and state the asymptotes.

Problem 14 Domain and hole from a graph ★★★

The graph is of f(x) = x² − x − 6x − 3. Identify the hole, give the domain, and write the equation of the line that the graph follows.

Problem 15 Solving for an input ★★★

For f(x) = 6x − 1 + 2, find the value of x for which f(x) = 5, and the value of x for which f(x) = 0.

Problem 16 A road trip ★★★

A road trip covers 240 miles, so the time in hours at a steady speed of r mph is t = 240r. Find the time at 60 mph, find the speed needed to finish in 3.2 hours, and describe what the graph does as r grows.

Problem 17 Diluting a solution ★★★

A beaker holds 10 liters of liquid that contains 2 liters of acid. After x liters of water are added, the fraction of acid is c(x) = 210 + x. How much water must be added so that the fraction is 0.1? Give the domain in this context and the horizontal asymptote, and interpret it.

Problem 18 Average cost of a food truck ★★★

A food truck has a $1,200 start-up cost and spends $3 on ingredients per meal. The average cost per meal after n meals is A(n) = 3n + 1,200n. Rewrite A(n) in the form 3 + kn, find n when the average cost is $5, and name the horizontal asymptote.

Problem 19 All the features of one function ★★★

For f(x) = x² − x − 12x² − 2x − 8, find the hole, the vertical asymptote, the horizontal asymptote, the x-intercept, and the y-intercept.

Problem 20 Designing a rational function ★★★

Write a rational function with a vertical asymptote at x = 4, a hole at x = −3, a horizontal asymptote y = 5, and an x-intercept at 7. Give the hole’s coordinates.