FunctionsAA HL

Reciprocal graphs:

Build the reciprocal graph from zeros, signs, key points and end behaviour.

  • Keep each input fixed and take the reciprocal of each non-zero output.
  • For a polynomial, each real zero gives a vertical asymptote of its reciprocal. Check the sign on each side.
  • Outputs and stay fixed. The reciprocal has the same sign as the original output.
  • If , then . Check end behaviour before stating a horizontal asymptote.

Example 1

Sketch , showing the asymptotes and turning point.

  1. Keep each input

    Take the reciprocal of each non-zero output.

  2. Find the asymptotes

    Set to locate the vertical asymptotes.

  3. Map the turning point

    The input stays the same. A negative minimum becomes a negative local maximum.

Original:
Original quadratic graph with its turning point and roots marked.
Overlay:
Dashed original quadratic and green reciprocal, with vertical and horizontal asymptotes where present.
Middle branch close-up:
Close-up of the negative middle reciprocal branch, showing its local maximum at minus 2, minus one quarter.
Original graphReciprocal graphAsymptotes
  • The roots and become vertical asymptotes; the reciprocal is undefined there.
  • The turning point maps to , a local maximum on the middle branch.
  • Outside the roots the reciprocal is positive; between them it is negative. It has no -intercepts.
  • As , . The horizontal asymptote is .
  • Approach from the left/right: / . Approach from the left/right: / .
Vertical asymptotes and Horizontal asymptote Turning point

Example 2

Sketch , showing the asymptotes and turning point.

  1. Check for zeros

    The original graph stays above the -axis, so no vertical asymptote is created.

  2. Map the key points

    Keep inputs and reciprocate outputs.

  3. Follow the ends

    Large positive outputs become small positive outputs.

Original:
Original quadratic graph with its turning point and roots marked.
Overlay:
Dashed original quadratic and green reciprocal, with vertical and horizontal asymptotes where present.
Original graphReciprocal graphAsymptotes
  • Since , there are no zeros and no vertical asymptotes.
  • The minimum becomes the maximum . The graph stays positive.
  • The horizontal asymptote is . The reciprocal range is .
  • Points with output or stay fixed under reciprocation. This graph has just one: .
No vertical asymptotesHorizontal asymptote Maximum

Remember
The reciprocal is different from the inverse function .

Your turn

5 practice questions
Question 1

For , sketch . State its asymptotes and turning point.

Check answer
Vertical asymptotes and Horizontal asymptote Local maximum
Original dashed quadratic and solid reciprocal answer graph.

Zeros become vertical asymptotes. Reciprocate the turning-point output; its input stays fixed.

Question 2

For , sketch . State its asymptotes and turning point.

Check answer
No vertical asymptotesHorizontal asymptote Maximum
Original dashed quadratic and solid reciprocal answer graph.

The original output is always positive, so there are no zeros. The positive minimum becomes a positive maximum.

Question 3

For , sketch . State its asymptotes and turning point.

Check answer
Vertical asymptotes and Horizontal asymptote Local maximum
Original dashed quadratic and solid reciprocal answer graph.

Zeros become vertical asymptotes. Reciprocate the turning-point output; its input stays fixed.

Question 4

For , sketch . State its asymptotes and turning point.

Check answer
No vertical asymptotesHorizontal asymptote Maximum
Original dashed quadratic and solid reciprocal answer graph.

The original output is always positive, so there are no zeros. The positive minimum becomes a positive maximum.

Question 5

For , sketch . State its asymptotes and turning point.

Check answer
Vertical asymptotes and Horizontal asymptote Local maximum
Original dashed quadratic and solid reciprocal answer graph.

Zeros become vertical asymptotes. Reciprocate the turning-point output; its input stays fixed.

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