1. TRANSLATE the problem information
- Given information:
- f is linear (so \(\mathrm{f(x) = mx + b}\))
- As x increases by 1, f(x) decreases by 5
- When \(\mathrm{x = -1}\), \(\mathrm{f(x) = 17}\)
- The key insight: "decreases by 5 for each increase of 1" means the slope is -5
2. INFER the approach
- Since we know f is linear, we need to find both m and b in \(\mathrm{f(x) = mx + b}\)
- We have the slope (\(\mathrm{m = -5}\)) and one point \(\mathrm{(-1, 17)}\)
- Strategy: Use point substitution to find b, then test our equation against each table
3. Set up the linear equation
- With slope \(\mathrm{m = -5}\): \(\mathrm{f(x) = -5x + b}\)
- We still need to find b
4. SIMPLIFY to find the y-intercept
- Use the known point \(\mathrm{(-1, 17)}\):
\(\mathrm{f(-1) = 17}\)
\(\mathrm{-5(-1) + b = 17}\)
\(\mathrm{5 + b = 17}\)
\(\mathrm{b = 12}\) - Complete equation: \(\mathrm{f(x) = -5x + 12}\)
5. SIMPLIFY by evaluating the function
- Test at \(\mathrm{x = -1}\): \(\mathrm{f(-1) = -5(-1) + 12 = 5 + 12 = 17}\) ✓
- Test at \(\mathrm{x = 0}\): \(\mathrm{f(0) = -5(0) + 12 = 12}\)
- Test at \(\mathrm{x = 2}\): \(\mathrm{f(2) = -5(2) + 12 = -10 + 12 = 2}\)
- Our function gives: \(\mathrm{(-1,17), (0,12), (2,2)}\)
6. INFER which table matches
- Choice (B) shows exactly these values: \(\mathrm{(-1,17), (0,12), (2,2)}\)
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret "decreases by 5" and think the slope is positive 5 instead of negative 5.
They reason: "The function changes by 5, so slope = 5." This leads them to write \(\mathrm{f(x) = 5x + b}\), find \(\mathrm{b = 12}\) using the point \(\mathrm{(-1,17)}\), getting \(\mathrm{f(x) = 5x + 12}\). When they evaluate this at \(\mathrm{x = 0}\) and \(\mathrm{x = 2}\), they get \(\mathrm{f(0) = 12}\) and \(\mathrm{f(2) = 22}\), which doesn't match any choice exactly, leading to confusion and guessing.
Second Most Common Error:
Inadequate SIMPLIFY execution: Students correctly identify slope = -5 and set up \(\mathrm{f(x) = -5x + b}\), but make arithmetic errors when solving for b or evaluating the function.
For example, when finding b: \(\mathrm{f(-1) = 17}\) becomes \(\mathrm{-5(-1) + b = 17}\), but they calculate \(\mathrm{-5(-1) = -5}\) instead of +5, leading to \(\mathrm{-5 + b = 17}\), so \(\mathrm{b = 22}\). This gives \(\mathrm{f(x) = -5x + 22}\), which when evaluated gives values that might lead them to select Choice (C).
The Bottom Line:
This problem tests whether students can correctly interpret rate language ("decreases by 5") as a negative slope, then systematically use point-slope methods to find the complete linear equation.
