The function f is defined by \(\mathrm{f(x) = (-8)(2)^x + 22}\). What is the y-intercept of the graph of \(\mathrm{y...

1. TRANSLATE the problem information

• We need to find the y-intercept of \(\mathrm{f(x) = (-8)(2)^x + 22}\)
• Y-intercept means the point where the graph crosses the y-axis
• This occurs when \(\mathrm{x = 0}\), so we need to find \(\mathrm{f(0)}\)

2. INFER the solution approach

• To find \(\mathrm{f(0)}\), substitute \(\mathrm{x = 0}\) into the function
• This will involve applying the zero exponent rule since we'll have \(\mathrm{2^0}\)

3. SIMPLIFY by substituting x = 0

• \(\mathrm{f(0) = (-8)(2)^0 + 22}\)
• Apply zero exponent rule: \(\mathrm{2^0 = 1}\)
• \(\mathrm{f(0) = (-8)(1) + 22}\)
• \(\mathrm{f(0) = -8 + 22 = 14}\)

4. TRANSLATE back to coordinate form

• Since \(\mathrm{f(0) = 14}\), the y-intercept is the point \(\mathrm{(0, 14)}\)

Answer: A. (0, 14)

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual gap with zero exponent rule: Students think that \(\mathrm{2^0 = 0}\) instead of \(\mathrm{2^0 = 1}\)

When they substitute: \(\mathrm{f(0) = (-8)(2)^0 + 22 = (-8)(0) + 22 = 0 + 22 = 22}\)

This leads them to select Choice C. \(\mathrm{(0, 22)}\)

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly find that \(\mathrm{f(0) = (-8)(1) + 22}\), but then ignore or forget the constant term +22

They calculate: \(\mathrm{f(0) = (-8)(1) = -8}\) and stop there

This may lead them to select Choice D. \(\mathrm{(0, -8)}\)

The Bottom Line:

The zero exponent rule is the key concept that trips up many students. Even if they know how to find y-intercepts, not remembering that any non-zero number to the power of 0 equals 1 will derail their entire solution.