Compare Differences with Ratios
Equal-step outputs can be inspected for constant differences or constant nonzero ratios.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: The outputs at inputs 0,1,2,3 are 2,4,8,16. Which rule describes them? Candidate answers: (A) y=2+2x (B) y=2x² (C) y=2·2^x (D) y=2·3^x.
- Independent check inputs: The outputs at inputs 0,1,2,3 are 3,6,12,24. Which rule describes them? Candidate answers: (A) y=3+2x (B) y=3x² (C) y=3·2^x (D) y=3·3^x.
Learn the idea
An exponential pattern changes by a constant multiplicative ratio over equal input steps. This differs from a linear pattern’s constant addition and can model supplied growth, decay or ordered geometric sequence terms. Equal-step outputs can be inspected for constant differences or constant nonzero ratios. A doubling table has a constant ratio while its differences grow, identifying exponential rather than linear structure.
Here is the method on the supplied teaching case. Consecutive outputs have constant ratio 2, while differences change. At x=0 the output is 2, so y=2·2^x matches every supplied pair.
A few values can match many unrestricted formulas, so compare the supplied candidate rules rather than claiming uniqueness among all mathematical functions. The initial value fixes the exponential coefficient here.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: The outputs at inputs 0,1,2,3 are 2,4,8,16. Which rule describes them? Candidate answers: (A) y=2+2x (B) y=2x² (C) y=2·2^x (D) y=2·3^x. Solution: Consecutive outputs have constant ratio 2, while differences change. At x=0 the output is 2, so y=2·2^x matches every supplied pair. The final answer is y=2·2^x. A few values can match many unrestricted formulas, so compare the supplied candidate rules rather than claiming uniqueness among all mathematical functions. The initial value fixes the exponential coefficient here.
Check your reasoning
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Helpful tips
- Equal-step outputs can be inspected for constant differences or constant nonzero ratios.
- A few values can match many unrestricted formulas, so compare the supplied candidate rules rather than claiming uniqueness among all mathematical functions.
See the answer and explanation
y=3·2^x
Consecutive outputs have constant ratio 2, while differences change. At x=0 the output is 3, so y=3·2^x matches every supplied pair.
Take it further
Changed-input task: The outputs at inputs 0,1,2,3 are 9,18,36,72. Which rule describes them? Candidate answers: (A) y=9+2x (B) y=9x² (C) y=9·2^x (D) y=9·3^x. Show the mathematical reason, not only the selected answer. Expected result: y=9·2^x. Consecutive outputs have constant ratio 2, while differences change. At x=0 the output is 9, so y=9·2^x matches every supplied pair.
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More practice: try new cases
Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.
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