Graphing Rational Functions

Explore vertical asymptotes, horizontal asymptotes, holes and intercepts.

Lesson 4 of 53 parts Practice Flashcards

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Vertical Asymptotes

Objective: Explain vertical asymptotes and use it to solve or interpret a problem.

A noncancelled denominator zero may produce a vertical asymptote; inspect behavior from both sides rather than calling every denominator zero an asymptote. A hole comes from a cancelled common factor.

Worked examples

Try itGraphing Rational Functions — practice 1.1: For y=4/(x−5)+3, what is the vertical asymptote x-value?Show answer

Step 1: The denominator is zero at x=5. Step 2: The numerator 4 remains nonzero. Step 3: The vertical asymptote is x=5.

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Vertical Asymptotes

Objective: Explain vertical asymptotes and use it to solve or interpret a problem.

Worked examples

Try itGraphing Rational Functions — practice 1.1: For y=4/(x−5)+3, what is the vertical asymptote x-value?Show answer

Step 1: The denominator is zero at x=5. Step 2: The numerator 4 remains nonzero. Step 3: The vertical asymptote is x=5.

Horizontal Asymptotes

Objective: Explain horizontal asymptotes and use it to solve or interpret a problem.

For f(x)=a/(x−h)+k with a≠0, output approaches k as |x| grows; an asymptote describes long-run behavior, not a wall the graph can never cross. No, other rational functions can cross one.

Worked examples

Try itGraphing Rational Functions — practice 2.1: For y=2/(x−8)+4, what value does y approach as x→+∞?Show answer

Step 1: The fraction 2/(x−8) tends to zero. Step 2: The remaining vertical shift is 4. Step 3: The horizontal asymptote is y=4.

Holes and Intercepts

Objective: Explain holes and intercepts and use it to solve or interpret a problem.

A cancelled factor leaves a missing point at its original excluded input; x-intercepts require an allowed input with numerator zero. Verify features from the uncancelled rule. No, the function has no output there.

Worked examples

Try itGraphing Rational Functions — practice 3.1: Simplify (x−3)(x+11)/(x−3) for x≠3. Evaluate at x=0.Show answer

Step 1: Cancel the shared factor only because x≠3. Step 2: The remaining expression is x+11. Step 3: At x=0, its value is 11.

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