Match Squared Inputs to a Table

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: At x=0,1,2,3 a rule has outputs 0,2,8,18. Which formula matches all four? Candidate answers: (A) y=2x (B) y=2x² (C) y=2x³ (D) y=x+2.
• Independent check inputs: At x=0,1,2,3 a rule has outputs 0,3,12,27. Which formula matches all four? Candidate answers: (A) y=3x (B) y=3x² (C) y=3x³ (D) y=x+3.

Learn the idea

A function assigns one output to each allowed input. A table can reveal nonlinear structure or interval-average change, while the domain records which inputs make sense mathematically and in the supplied context. A quadratic table can arise by multiplying squared inputs by a constant. Its equal-input-step output differences change, which distinguishes its pattern from a linear relationship.

Here is the method on the supplied teaching case. The values are 2 times 0²,1²,2²,3². First differences are 2,6,10, not constant; this is not a linear rule.

Matching just one point cannot identify a rule. Check all supplied rows and compare first differences; the shown table supports the candidate formula among the stated options, not every conceivable formula outside the data.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: At x=0,1,2,3 a rule has outputs 0,9,36,81. Which formula matches all four? Candidate answers: (A) y=9x (B) y=9x² (C) y=9x³ (D) y=x+9. Show the mathematical reason, not only the selected answer. Expected result: y=9x². The values are 9 times 0²,1²,2²,3². First differences are 9,27,45, not constant; this is not a linear rule.

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