Graphing Rational Functions

Develop vertical asymptotes, horizontal asymptotes, holes and intercepts.

Lesson 4 of 43 parts Practice Flashcards

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Vertical Asymptotes

Objective: Explain vertical asymptotes, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

For y=A/(x−h)+k, A≠0: vertical asymptote x=h

Worked examples

Try itFor y=3/(x−7)+5, what is the vertical asymptote x-value?Show answer

Step 1: The denominator is zero at x=7. Step 2: The numerator 3 remains nonzero. Step 3: The vertical asymptote is x=7.

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Vertical Asymptotes

Objective: Explain vertical asymptotes, apply the rule, and justify relevant restrictions.

Formulas

For y=A/(x−h)+k, A≠0: vertical asymptote x=h

Worked examples

Try itFor y=3/(x−7)+5, what is the vertical asymptote x-value?Show answer

Step 1: The denominator is zero at x=7. Step 2: The numerator 3 remains nonzero. Step 3: The vertical asymptote is x=7.

Horizontal Asymptotes

Objective: Explain horizontal asymptotes, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

For y=A/(x−h)+k: horizontal asymptote y=k

Worked examples

Try itFor y=3/(x−7)+5, what value does y approach as x→+∞?Show answer

Step 1: The fraction 3/(x−7) tends to zero. Step 2: The remaining vertical shift is 5. Step 3: The horizontal asymptote is y=5.

Holes and Intercepts

Objective: Explain holes and intercepts, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$(x−a)g(x)/(x−a)=g(x) for x≠a$

Worked examples

Try itSimplify (x−3)(x+7)/(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+7. Step 3: At x=0, its value is 7.

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