Problem

For the secant through \((1,~2)\) and \((5,~10)\), calculate the average rate of change, write the secant-line equation, and interpret the signed rate per input unit.

Prompt graph connecting the ordered pairs (1,2) and (5,10), with both endpoint labels fully visible and the slope withheld.

Big Picture

What this problem is really about

The average rate and the secant line are two views of the same slope. Compute rise over run from the two plotted points using consistent subtraction, then pair that slope with either endpoint in point-slope form. Simplifying the line provides a built-in check that both sourced points satisfy the equation.

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Calculation: (2−10)/(5−1) = −2. Line: y−2 = −2(x−1), or y = −2x+4. Interpretation: output decreases by 2 per input unit.

Calculation: (10−2)/(5−1) = 2. Line: y−2 = 2(x−1), or y = 2x. Interpretation: output increases by 2 per input unit.

Calculation: (5−1)/(10−2) = 1/2. Line: y−2 = (1/2)(x−1). Interpretation: output increases by 1/2 per input unit.

Calculation: (10−2)/5 = 8/5. Line: y−2 = (8/5)(x−1). Interpretation: output increases by 8/5 per input unit.

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For the secant through \((1,~2)\) and \((5,~10)\), calculate the average rate of change, write the secant-line equation, and interpret the signed rate per input unit.

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