A linear function t satisfies the following: The average rate of change of t between x = -2 and x...

1. TRANSLATE the problem information

• Given information:
  • Average rate of change of t between \(\mathrm{x = -2}\) and \(\mathrm{x = 3}\) is -5
  • \(\mathrm{t(3) = 17}\)
  • Need to find the equation of linear function t

2. INFER the key insight about linear functions

• For any linear function, the average rate of change over any interval equals the slope
• This means: slope \(\mathrm{m = -5}\)

3. INFER how to find the complete equation

• We have slope \(\mathrm{m = -5}\) and point \(\mathrm{(3, 17)}\)
• Use the linear function form \(\mathrm{t(x) = mx + b}\) to find b

4. SIMPLIFY to find the y-intercept

• Substitute the known point into \(\mathrm{t(x) = mx + b}\):
  • \(\mathrm{t(3) = 17}\), so: \(\mathrm{17 = -5(3) + b}\)
  • \(\mathrm{17 = -15 + b}\)
  • \(\mathrm{b = 32}\)

5. Write the final equation

• \(\mathrm{t(x) = -5x + 32}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students don't recognize that "average rate of change" for a linear function equals the slope. They might try to calculate it using the rate of change formula: \(\mathrm{\frac{t(3) - t(-2)}{3 - (-2)}}\), but they don't know \(\mathrm{t(-2)}\), creating confusion.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that slope = -5 and try to use point \(\mathrm{(3, 17)}\), but make algebraic errors when solving \(\mathrm{17 = -5(3) + b}\). Common mistakes include:

• Getting \(\mathrm{b = 2}\) instead of \(\mathrm{b = 32}\) (arithmetic error: \(\mathrm{17 - (-15) = 17 + 15 = 32}\), not 2)
• Forgetting the negative sign: calculating \(\mathrm{17 = 5(3) + b}\) instead

These errors may lead them to select Choice C (\(\mathrm{t(x) = -5x + 17}\)) or other incorrect options.

The Bottom Line:

This problem tests whether students understand that linear functions have constant rate of change (the slope) and can use point-slope relationships effectively. The key insight is recognizing what "average rate of change" means for linear functions specifically.