Algebra 2 Rational Functions Worksheet with Answers – Printable

Algebra 2 Rational Functions Worksheet: problems 1–5

1. Domain of a rational function

State the domain of each function.

1. \(f(x)=\frac{3}{x+2}\)
2. \(g(x)=\frac{1}{x-7}\)
3. \(h(x)=\frac{x}{2x-6}\)
4. \(k(x)=\frac{x+1}{x^{2}-25}\)

2. Evaluate a rational function

Let \(f(x)=\frac{x+1}{x-2}\). Find \(f(-1)\), \(f(0)\), \(f(1)\), \(f(3)\) and \(f(5)\).

3. Table for y = 6/x

Copy and complete the table for \(y=\frac{6}{x}\).

\(x\)	\(-6\)	\(-3\)	\(-2\)	\(-1\)	\(1\)	\(2\)	\(3\)	\(6\)
\(y\)

4. Vertical asymptotes

Find every vertical asymptote.

1. \(y=\frac{4}{x+3}\)
2. \(y=\frac{2x}{x-1}\)
3. \(y=\frac{1}{x^{2}-4}\)
4. \(y=\frac{x+5}{x^{2}+3x}\)

5. Horizontal asymptotes

Find the horizontal asymptote of each function, or write none.

1. \(y=\frac{3x+1}{x-4}\)
2. \(y=\frac{5}{x^{2}+1}\)
3. \(y=\frac{2x^{2}-1}{3x^{2}+x}\)
4. \(y=\frac{x^{2}+1}{x-1}\)
5. \(y=\frac{-4x}{2x+7}\)

Algebra 2 Rational Functions Worksheet: problems 6–10

6. Intercepts

Find the x-intercept and the y-intercept of each function.

1. \(f(x)=\frac{2x-6}{x+1}\)
2. \(g(x)=\frac{x+4}{x-2}\)

7. Find the hole

Factor, cancel, and give the coordinates of the hole in each graph.

1. \(f(x)=\frac{x^{2}-9}{x-3}\)
2. \(g(x)=\frac{x^{2}+x-6}{x-2}\)
3. \(h(x)=\frac{(x-4)(x+1)}{(x-4)(x+2)}\)

8. Hole, asymptote and end behavior

Let \(f(x)=\frac{x^{2}-4x+3}{x^{2}-1}\). Find the hole, the vertical asymptote and the horizontal asymptote.

9. Shifts of y = 1/x

Describe how each graph is related to \(y=\frac{1}{x}\), then name its asymptotes.

1. \(y=\frac{1}{x+3}-2\)
2. \(y=-\frac{1}{x-5}+4\)
3. \(y=\frac{1}{x}+6\)

10. Read a hyperbola

The graph of a rational function is shown.

1. Name the vertical and horizontal asymptotes.
2. State the domain and the range.
3. Write the equation in the form \(y=\frac{a}{x-h}+k\).

Algebra 2 Rational Functions Worksheet: problems 11–15

11. Graph a rational function

Graph \(y=\frac{2}{x+1}-3\) on a coordinate plane. Draw the asymptotes as dashed lines and plot at least four points.

12. Equation from a graph

Write an equation for the graph shown.

13. A graph with a hole

The graph of \(f(x)=\frac{x^{2}-1}{x-1}\) is shown.

1. Simplify \(f(x)\) for \(x\neq 1\).
2. Give the coordinates of the open circle.
3. State the domain.

14. Write a rational function

Write a function of the form \(y=\frac{a}{x-h}+k\) with vertical asymptote \(x=2\), horizontal asymptote \(y=-1\) and passing through \((3,1)\).

15. Inverse variation constant

The variable \(y\) varies inversely with \(x\), and \(y=8\) when \(x=5\).

1. Find the constant \(k\) and write the equation.
2. Find \(y\) when \(x=20\).
3. Find \(x\) when \(y=16\).

Algebra 2 Rational Functions Worksheet: problems 16–20

16. Is it inverse variation?

Decide whether each table shows inverse variation. If it does, find \(k\).

1. \((2,18),(3,12),(4,9),(6,6)\)
2. \((1,5),(2,10),(3,15),(4,20)\)
3. \((5,12),(10,6),(15,4),(20,3)\)

17. Inverse variation from a graph

The graph shows an inverse variation between \(x\) and \(y\).

1. Find \(k\) and write the equation.
2. Find \(y\) when \(x=16\).

18. Road trip

Jamal’s family drives 240 miles to a lake. The trip takes \(t=\frac{240}{r}\) hours at an average speed of \(r\) miles per hour.

1. Find \(t\) for \(r=40,48,60\) and \(80\).
2. What speed makes the trip take 2.5 hours?
3. What is the horizontal asymptote, and what does it mean here?

19. Cookie cost

Maya bakes cookies for a bake sale. The ingredients and pans cost $30, and each cookie costs $0.50 more to make. The average cost per cookie for \(n\) cookies is \(C(n)=\frac{0.5n+30}{n}\) dollars.

1. Find \(C(20)\), \(C(60)\) and \(C(120)\).
2. Rewrite \(C(n)\) as \(0.5+\frac{30}{n}\) and name the horizontal asymptote.
3. How many cookies must Maya bake for the average cost to be $0.60?

20. Bus rental

A class trip needs a bus that costs $180. The cost per student is \(c=\frac{180}{n}\) for \(n\) students.

1. Find the cost per student when 12 students go and when 20 students go.
2. How many students are needed to bring the cost down to $6 each?

Algebra 2 Rational Functions Worksheet: problems 21–22

21. Bike pump

Priya’s bike pump holds 90 cubic inches of air at a pressure of 15 pounds per square inch (psi). Pressure and volume vary inversely.

1. Write an equation for pressure \(P\) in terms of volume \(V\).
2. Find the pressure when the air is squeezed to 54 cubic inches.
3. Find the volume at 45 psi.

22. Juice mixture

Ethan has 10 gallons of juice drink that is 20% orange concentrate, so it holds 2 gallons of concentrate. He adds \(x\) gallons of pure concentrate. The fraction of concentrate in the mixture is \(c(x)=\frac{2+x}{10+x}\).

1. Find \(c(0)\), \(c(10)\) and \(c(30)\) as percents.
2. How many gallons must Ethan add to make the mixture 50% concentrate?
3. Name the horizontal asymptote and explain why \(c(x)\) can never reach it.

How do the early problems work?

Problem 1 asks for the domain of \(\frac{x}{2x-6}\), where the bottom vanishes at 3. Problem 2 evaluates \(f(x)=\frac{x+1}{x-2}\) at five inputs, so \(f(3)=4\) and \(f(5)=2\). Then the table for \(y=\frac{6}{x}\) shows values shrinking as \(x\) grows. Problem 5 asks for horizontal asymptotes, such as \(y=3\) for \(\frac{3x+1}{x-4}\).

What should you know about holes?

A hole appears when a factor cancels. In problem 7, \(\frac{x^{2}-9}{x-3}\) reduces to \(x+3\), so the hole sits at \((3,6)\). Problem 8 combines everything for \(\frac{x^{2}-4x+3}{x^{2}-1}\): the hole at \(x=1\), a vertical asymptote at \(x=-1\) and a horizontal asymptote at \(y=1\). Find the hole’s height from the simplified formula, not the original.

Which applications need care?

Problem 15 gives \(y=8\) at \(x=5\), so \(k=40\) and \(y=\frac{40}{x}\). Problem 16 tests tables: \((2,18)\), \((3,12)\), \((4,9)\) and \((6,6)\) all multiply to 36. The road trip in problem 18 has \(t=\frac{240}{r}\), so a speed of 60 mph gives 4 hours. In the bike pump problem, the product of pressure and volume stays 1,350. Finally, the juice mixture problem builds a concentration that approaches a limit as pure water is added.

How can you finish this worksheet well?

Choose two graphs and describe them in a sentence each, naming the asymptotes and the intercepts. Next, compare with the key. If the sentence and the picture disagree, trust the algebra and redraw. Afterward, take the quiz for a fast check.

See every chapter on the Algebra 2 math page.