Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Determine both the initial factor and multiplicative growth factor.
- Justify the conclusion "b=2, so f(t)=3·2ᵗ" using the stated assumptions.
Before you start
Exponents and positive roots.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
For f(t)=abᵗ with a,b>0, use f(0)=3 and f(2)=12.
Why this math matters
Determine both the initial factor and multiplicative growth factor. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The exponential form is assumed throughout the observed interval.
- The base and initial factor are positive.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Recover an exponential model from two readings
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For f(t)=abᵗ with a,b>0, use f(0)=3 and f(2)=12.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
f(0)=a=3
At zero the exponential factor equals one.
Work through the mathematics
3b²=12 ⇒ b²=4
The second reading identifies the two-step multiplier.
Check the conclusion
b=2, so f(t)=3·2ᵗ
Positivity of the base selects two; one-step and two-step growth must not be confused.
The result
b=2, so f(t)=3·2ᵗ
Positivity of the base selects two; one-step and two-step growth must not be confused.
Common mistakes to catch
- Two readings fit a model but do not establish its validity beyond those readings.
- State the time unit associated with the growth factor.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is f(1)?
Show a hint
Use the calibrated model.
Reveal answer and explanation
Six
3·2=6.
Practice 2
Does increasing from three to twelve mean b=4?
Show a hint
The change occurred over two time units.
Reveal answer and explanation
No
Four is b², while b=2 is the one-unit multiplier.
Take the idea with you
Test a third reading before extending a calibrated exponential trend.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Add repeated multiplicative contributions efficiently
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