Learn with Amar
Teaching video
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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Separate power-law fitting from exponential fitting.
- Justify the conclusion "y=3x² and log y=log3+2log x" using the stated assumptions.
Before you start
Logarithm laws and slopes.
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
If y=kxᵃ for x>0 and y(2)=12,y(4)=48, find k and a.
Why this math matters
Separate power-law fitting from exponential fitting. 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:
- A positive power-law model is assumed.
- Both observations use the same measurement units.
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.
Recognize a power law on logarithmic axes
PausedQuestion: Start with the question. Paused.
Question
Start with the question
If y=kxᵃ for x>0 and y(2)=12,y(4)=48, find k and a.
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
48/12=(4/2)ᵃ gives 4=2ᵃ
Taking a ratio removes the unknown multiplicative constant.
Work through the mathematics
a=2 and 12=k·2² gives k=3
The two observations determine the assumed power model.
Check the conclusion
y=3x² and log y=log3+2log x
A log-log plot has slope two; an ordinary time-exponential model would use different axes.
The result
y=3x² and log y=log3+2log x
A log-log plot has slope two; an ordinary time-exponential model would use different axes.
Common mistakes to catch
- A straight semilog plot and a straight log-log plot indicate different model forms.
- Two fitted observations do not validate unlimited extrapolation.
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 does doubling x do in this model?
Show a hint
Use the power exponent.
Reveal answer and explanation
It multiplies y by four
(2x)²=4x².
Practice 2
Why require x and y positive for this log plot?
Show a hint
Real logarithms need positive inputs.
Reveal answer and explanation
Otherwise the displayed logarithms are undefined
A different transformation would need separate assumptions.
Take the idea with you
Choose axes that turn the proposed model, rather than a different model, into a straight line.
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: Find the line a rational graph approaches
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