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Grade 11 · Exponential calibration

Exponential calibration: Moderate decay from a larger base

Exponential calibration: investigate moderate decay from a larger base with initial value a = 2.5; exponential rate b = -0.2.

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

Watch the relationship

Paused
Exponential calibration: Moderate decay from a larger base. Time / input: 0. Current amount: 2.5. Instantaneous rate: -0.5. Growth multiplier: 1A constant proportional rate02.5024yx → · labeled axes rescale to this model
This is an ideal exponential model with constant rate, positive initial value, and arbitrary time units. It does not include resource limits or measurement noise. Playback changes time while holding the parameters fixed.

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.

Time / input
0
Current amount
2.5
Instantaneous rate
-0.5
Growth multiplier
1

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From experiment to screen.

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Understand what you are seeing

The idea behind the motion.

The initial amount and proportional growth rate play different roles. Multiplying a scales every output, but changing b changes each time interval's multiplicative factor. A negative rate produces decay without making a positive starting amount negative. This investigation starts with Initial value a = 2.5; Exponential rate b = -0.2. 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(t)=a e^(bt); f′(t)=b f(t)

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

    Read the initial value at t=0 and predict whether the rate produces growth, decay, or a constant curve. The starting case is “Moderate decay from a larger base.”

  2. STEP 2

    Follow the changing quantity

    Trace four time units. Compare the changing amount with its instantaneous rate b times the amount.

  3. STEP 3

    Explain and test the result

    Divide the ending amount by the starting amount to recover e^(bt). Compare equal time intervals rather than equal changes in height.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Initial value a = 2.5; Exponential rate b = -0.2. Pause the timeline at 100%. Given time / input = 4, calculate current amount, instantaneous rate, growth multiplier. 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

Substitution gives y=2.5 exp((-0.2)(4))=1.123322. The rate is k y=(-0.2)(1.123322)=-0.224664, rather than just the amount y. Results: Current amount: 1.123; Instantaneous rate: -0.225; Growth multiplier: 0.449. 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

Recover an exponential model from two readings

Determine both the initial factor and multiplicative growth factor.