For the exponential function f, the value of \(\mathrm{f(0)}\) is c, where c is a constant. Of the following equations...

1. TRANSLATE the problem requirements

• Given information:
  • \(\mathrm{f(0) = c}\) (where c is a constant)
  • Need to find which equation shows c as the coefficient or the base

• What this tells us: First find the value of c by evaluating \(\mathrm{f(0)}\), then check which equation displays that value prominently

2. SIMPLIFY by evaluating f(0) for each choice

Choice A: \(\mathrm{f(x) = 22(1.5)^{(x+1)}}\)

• \(\mathrm{f(0) = 22(1.5)^{(0+1)}}\)
• \(\mathrm{= 22(1.5)^1}\)
• \(\mathrm{= 33}\)

Choice B: \(\mathrm{f(x) = 33(1.5)^x}\)

• \(\mathrm{f(0) = 33(1.5)^0}\)
• \(\mathrm{= 33(1)}\)
• \(\mathrm{= 33}\)

Choice C: \(\mathrm{f(x) = 49.5(1.5)^{(x-1)}}\)

• \(\mathrm{f(0) = 49.5(1.5)^{(-1)}}\)
• \(\mathrm{= 49.5 \div 1.5}\)
• \(\mathrm{= 33}\)

Choice D: \(\mathrm{f(x) = 74.25(1.5)^{(x-2)}}\)

• \(\mathrm{f(0) = 74.25(1.5)^{(-2)}}\)
• \(\mathrm{= 74.25 \div 2.25}\)
• \(\mathrm{= 33}\)

So \(\mathrm{c = 33}\) for all functions.

3. INFER which equation shows c as coefficient or base

Now that we know \(\mathrm{c = 33}\), let's check each equation:

• Choice A: coefficient = 22, base = 1.5 → neither equals 33
• Choice B: coefficient = 33, base = 1.5 → coefficient equals c!
• Choice C: coefficient = 49.5, base = 1.5 → neither equals 33
• Choice D: coefficient = 74.25, base = 1.5 → neither equals 33

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students evaluate \(\mathrm{f(0)}\) correctly but fail to connect this back to the question's requirement about showing c "as the coefficient or the base."

They might think: "All functions give \(\mathrm{f(0) = 33}\), so they're all equivalent" without realizing they need to identify which equation explicitly displays the value 33 in its structure. This leads to confusion and guessing among the choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make calculation errors with negative exponents in choices C and D.

For example, they might incorrectly evaluate \(\mathrm{49.5(1.5)^{(-1)}}\) as \(\mathrm{49.5 \times 1.5 = 74.25}\) instead of \(\mathrm{49.5 \div 1.5 = 33}\). This makes them think different choices give different values of \(\mathrm{f(0)}\), leading them to select the wrong answer based on faulty arithmetic.

The Bottom Line:

This problem tests whether students understand that equivalent exponential functions can be written in different forms, and they must identify which form makes a specific value (the y-intercept) most visible in the equation's structure.