Function f is defined by \(\mathrm{f(x) = -a^x + b}\), where a and b are constants. In the xy-plane, the...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = -a^x + b}\) (exponential function)
  • Graph of \(\mathrm{y = f(x) - 12}\) has y-intercept at \(\mathrm{(0, -75/7)}\)
  • Product of a and b equals \(\mathrm{320/7}\)
• What this tells us: We need to find where the modified function \(\mathrm{f(x) - 12}\) crosses the y-axis

2. INFER the approach

• At any y-intercept, \(\mathrm{x = 0}\)
• We need to evaluate \(\mathrm{f(0) - 12}\) and set it equal to \(\mathrm{-75/7}\)
• This will give us one equation to find either a or b

3. SIMPLIFY the function evaluation

• Start with: \(\mathrm{f(x) - 12 = (-a^x + b) - 12 = -a^x + b - 12}\)
• At \(\mathrm{x = 0}\): \(\mathrm{f(0) - 12 = -a^0 + b - 12}\)
• Since \(\mathrm{a^0 = 1}\): \(\mathrm{f(0) - 12 = -1 + b - 12 = b - 13}\)

4. TRANSLATE the y-intercept condition

• The y-intercept \(\mathrm{(0, -75/7)}\) means: when \(\mathrm{x = 0}\), \(\mathrm{y = -75/7}\)
• Therefore: \(\mathrm{b - 13 = -75/7}\)

5. SIMPLIFY to find b

• \(\mathrm{b - 13 = -75/7}\)
• \(\mathrm{b = -75/7 + 13}\)
• \(\mathrm{b = -75/7 + 91/7}\)
• \(\mathrm{b = 16/7}\)

6. INFER how to use the product constraint

• We know \(\mathrm{ab = 320/7}\) and \(\mathrm{b = 16/7}\)
• We can substitute to find a

7. SIMPLIFY to find a

• \(\mathrm{a \times (16/7) = 320/7}\)
• \(\mathrm{a = (320/7) \div (16/7)}\)
• \(\mathrm{a = (320/7) \times (7/16)}\)
• \(\mathrm{a = 320/16}\)
• \(\mathrm{a = 20}\)

Answer: 20

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often misinterpret what "the graph of \(\mathrm{y = f(x) - 12}\)" means, thinking they need to find the y-intercept of \(\mathrm{f(x)}\) instead of \(\mathrm{f(x) - 12}\).

They might calculate \(\mathrm{f(0) = -a^0 + b = b - 1}\) and set this equal to \(\mathrm{-75/7}\), leading to \(\mathrm{b = -75/7 + 1 = -68/7}\). Then using \(\mathrm{ab = 320/7}\), they get \(\mathrm{a = (320/7) \times (7/-68) = -320/68 \approx -4.7}\), which doesn't match any reasonable answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{b - 13 = -75/7}\) but make arithmetic errors when adding fractions.

A common mistake is \(\mathrm{b = -75/7 + 13 = -75/20}\) (incorrectly treating 13 as 13/20 instead of 91/7). This leads to wrong values throughout the rest of the solution and ultimately incorrect answer selection.

The Bottom Line:

This problem tests whether students can carefully track function transformations and handle fraction arithmetic accurately. The key insight is recognizing that \(\mathrm{f(x) - 12}\) creates a vertical shift that affects the y-intercept calculation.