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 \(\mathrm{y = f(x) - 15}\) has y-intercept at \(\mathrm{(0, -99/7)}\)
  • Product constraint: \(\mathrm{ab = 65/7}\)
• Need to find: value of a

2. TRANSLATE the y-intercept condition

• The graph \(\mathrm{y = f(x) - 15 = -a^x + b - 15}\) passes through \(\mathrm{(0, -99/7)}\)
• At \(\mathrm{x = 0}\): \(\mathrm{y = -a^0 + b - 15}\)

3. INFER the strategy

• Since \(\mathrm{a^0 = 1}\) for any nonzero a, we can find b first
• Then use the constraint \(\mathrm{ab = 65/7}\) to find a

4. SIMPLIFY to find b

• Substitute the y-intercept: \(\mathrm{-99/7 = -1 + b - 15}\)
• Combine constants: \(\mathrm{-99/7 = b - 16}\)
• Solve for b: \(\mathrm{b = -99/7 + 16 = -99/7 + 112/7 = 13/7}\)

5. SIMPLIFY using the product constraint

• From \(\mathrm{ab = 65/7}\) and \(\mathrm{b = 13/7}\):
• \(\mathrm{a(13/7) = 65/7}\)
• Divide both sides by 13/7: \(\mathrm{a = (65/7) ÷ (13/7) = 65/13 = 5}\)

Answer: 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about exponential functions: Students might not remember that \(\mathrm{a^0 = 1}\), treating \(\mathrm{a^0}\) as 0 instead.

If they use \(\mathrm{a^0 = 0}\), they get:

\(\mathrm{-99/7 = 0 + b - 15 = b - 15}\)

So \(\mathrm{b = -99/7 + 15 = -99/7 + 105/7 = 6/7}\)

Then using \(\mathrm{ab = 65/7}\): \(\mathrm{a(6/7) = 65/7}\), giving \(\mathrm{a = 65/6 ≈ 10.83}\)

This leads to confusion since this isn't among typical answer choices, causing them to guess.

Second Most Common Error:

Poor SIMPLIFY execution with fraction arithmetic: Students make calculation errors when adding fractions or dividing by fractions.

For example, they might incorrectly calculate:

\(\mathrm{-99/7 + 16}\) as \(\mathrm{-99/23}\) instead of converting 16 to 112/7

This leads to wrong values for b and subsequently wrong values for a.

The Bottom Line:

This problem tests both conceptual knowledge of exponential functions (particularly \(\mathrm{a^0 = 1}\)) and careful algebraic manipulation with fractions. Success requires systematic work through the constraint equations rather than trying to work backwards from answer choices.