FunctionsAI HL

Graphing a given reciprocal function - using a graphing calculator

Plot a supplied reciprocal function on a graphing calculator, then explain its key features.

  • Enter the given function as , then enter . Put the whole denominator in brackets if entering its formula directly.
  • Choose a window that shows the branches. Use the table, trace and minimum/maximum tools to investigate points; widen the window to check end behaviour.
  • At a defined point with , the same input gives . The -coordinate stays the same; only the -coordinate changes.
  • For the continuous polynomial denominators in this card, an -intercept of gives a vertical asymptote of . The reciprocal is undefined there. A -intercept does not become an asymptote.
  • Outputs and stay fixed. The reciprocal keeps the sign of each non-zero output and has no -intercepts. If , then .

Example 1

Use a GDC to plot the given function and sketch , showing the asymptotes and turning point.

  1. Plot on your GDC

    Enter the supplied denominator as Y1 and its reciprocal as Y2. Adjust the window and use a table to check each branch.

  2. Keep each input

    Take the reciprocal of each non-zero output.

  3. Find the asymptotes

    Use the zero or intersection tool on Y1. Check the Y2 table near each zero; the reciprocal is undefined at that input.

  4. Locate the turning point on your GDC

    Use the maximum tool on Y2. Its -coordinate matches the turning point of Y1; compare the two outputs in the table.

  5. Check the features

    Use trace or the minimum/maximum tool to confirm the marked turning point. Check values close to each excluded input on both sides.

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

Use a GDC to plot the given function and sketch , showing the asymptotes and turning point.

  1. Plot on your GDC

    Enter the supplied denominator as Y1 and its reciprocal as Y2. Adjust the window and use a table to check each branch.

  2. Check for zeros

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

  3. Map the key points

    Keep inputs and reciprocate outputs.

  4. Follow the ends

    Large positive outputs become small positive outputs.

  5. Check the features

    Use trace or the minimum/maximum tool to confirm the marked turning point. Check values close to each excluded input on both sides.

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
This is calculator graphing of a stated function. The general transformation is an explicit AA HL topic, not a separate AI HL transformation requirement. A calculator can draw a misleading line across an asymptote: check the denominator and nearby table values. Also, is not the inverse function .

Your turn

Question 1

For , use a GDC to plot and sketch . State its asymptotes and turning point.

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

Enter the stated formula as Y1 and 1/Y1 as Y2. Adjust the window and use the table or trace. Zeros become vertical asymptotes. Reciprocate the turning-point output; its input stays fixed. Check nearby values on both sides of every zero of the denominator; do not join branches across an excluded input.

Question 2

For , use a GDC to plot and sketch . State its asymptotes and turning point.

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

Enter the stated formula as Y1 and 1/Y1 as Y2. Adjust the window and use the table or trace. The original output is always positive, so there are no zeros. The positive minimum becomes a positive maximum. Check nearby values on both sides of every zero of the denominator; do not join branches across an excluded input.

Question 3

For , use a GDC to plot and sketch . State its asymptotes and turning point.

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

Enter the stated formula as Y1 and 1/Y1 as Y2. Adjust the window and use the table or trace. Zeros become vertical asymptotes. Reciprocate the turning-point output; its input stays fixed. Check nearby values on both sides of every zero of the denominator; do not join branches across an excluded input.

Question 4

For , use a GDC to plot and sketch . State its asymptotes and turning point.

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

Enter the stated formula as Y1 and 1/Y1 as Y2. Adjust the window and use the table or trace. The original output is always positive, so there are no zeros. The positive minimum becomes a positive maximum. Check nearby values on both sides of every zero of the denominator; do not join branches across an excluded input.

Question 5

For , use a GDC to plot and sketch . State its asymptotes and turning point.

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

Enter the stated formula as Y1 and 1/Y1 as Y2. Adjust the window and use the table or trace. Zeros become vertical asymptotes. Reciprocate the turning-point output; its input stays fixed. Check nearby values on both sides of every zero of the denominator; do not join branches across an excluded input.

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