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Grade 11 · Rational functions

Rational functions: A reciprocal centered at zero

Rational functions: investigate a reciprocal centered at zero with numerator a = 1; pole location b = 0.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Rational functions: A reciprocal centered at zero. Input x: 0.25. Reciprocal value: 4. Slope: -16. Excluded input: 0Two branches. One excluded input.-44-404yx → · labeled axes rescale to this model
a is positive. The two branches are drawn separately, with a gap around the excluded pole; the moving point stays strictly on its right. Finite plotted segments do not include infinity or establish a numerical value at the pole.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Input x
0.25
Reciprocal value
4
Slope
-16
Excluded input
0

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Understand what you are seeing

The idea behind the motion.

A shifted reciprocal has an excluded input at x=b and approaches the horizontal axis far from that input. A positive numerator makes the right branch positive and decreasing. Following points at known distances from the pole separates a large function value from a genuinely undefined input. This investigation starts with Numerator a = 1; Pole location b = 0. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

f(x)=a/(x−b); f′(x)=−a/(x−b)²

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Mark the vertical asymptote x=b and keep it out of the function's domain. The starting case is “A reciprocal centered at zero.”

  2. STEP 2

    Follow the changing quantity

    Follow the right-hand branch from distance 0.25 to distance 4 from the pole. Relate each value to the reciprocal of that distance.

  3. STEP 3

    Explain and test the result

    Change the numerator and compare vertical scaling while the pole stays fixed. No finite point on this branch reaches zero.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Numerator a = 1; Pole location b = 0. Pause the timeline at 20%. Given input x = 1, calculate reciprocal value, slope, excluded input. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

The denominator is 1−(0)=1, so 1/(1)=1. Its derivative is −1/(1)²=-1. Results: Reciprocal value: 1; Slope: -1; Excluded input: 0. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 11 · Algebra

Expose a rational graph's shifted reciprocal structure

Rewrite a rational function to identify asymptotes and range restrictions.