Graphing Rational Functions
A free Precalculus lesson from the “Rational Functions” unit, with a worked example and practice problems including step-by-step solutions.
A rational graph sketch is built from restrictions, intercepts, asymptotes, and interval signs. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.
What you'll learn
- Combine intercepts, holes, asymptotes, and sign checks to sketch rational graphs
- Use graphing rational functions in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. To sketch a rational function, which list is most useful?
- Rational graphs have several landmarks.
- Restrictions and asymptotes shape the graph.
- Intercepts and sign checks place branches.
Answer: holes, vertical asymptotes, intercepts, and end behavior
Practice problems
1. To sketch a rational function, which list is most useful?
Choices: holes, vertical asymptotes, intercepts, and end behavior · only the y-intercept · only the leading coefficient · only the largest exponent
Show solution
- Rational graphs have several landmarks.
- Restrictions and asymptotes shape the graph.
- Intercepts and sign checks place branches.
Answer: holes, vertical asymptotes, intercepts, and end behavior
2. To sketch a rational function, which list is most useful? (variation 2)
Choices: holes, vertical asymptotes, intercepts, and end behavior · only the y-intercept · only the leading coefficient · only the largest exponent
Show solution
- Rational graphs have several landmarks.
- Restrictions and asymptotes shape the graph.
- Intercepts and sign checks place branches.
Answer: holes, vertical asymptotes, intercepts, and end behavior
3. For r(x) = 1/(x - 2), the graph has branches near:
Choices: x = 2 · y = 2 only · x = 0 only · x = -2 and x = 2 as holes
Show solution
- Core Practice: First identify exactly what the question is asking: For r(x) = 1/(x - 2), the graph has branches near:
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- The denominator is zero at x = 2.
- That creates a vertical asymptote.
- Branches approach that line.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: x = 2
4. For r(x) = 1/(x - 2), the graph has branches near: (variation 2)
Choices: x = 2 · y = 2 only · x = 0 only · x = -2 and x = 2 as holes
Show solution
- Core Practice: First identify exactly what the question is asking: For r(x) = 1/(x - 2), the graph has branches near:
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- The denominator is zero at x = 2.
- That creates a vertical asymptote.
- Branches approach that line.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: x = 2
5. A sign chart for a rational graph helps decide:
Choices: whether each interval is above or below the x-axis · the exact font size of labels · the course order · the value of pi
Show solution
- Intervals are split by zeros and restrictions.
- Testing signs tells whether the graph is positive or negative.
- That helps place branches.
Answer: whether each interval is above or below the x-axis
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