The function g is defined by \(\mathrm{g(x) = a(4)^x + 10}\), where a is a constant. The graph of \(\mathrm{y...

1. TRANSLATE the y-intercept information

• Given information:
  • Function: \(\mathrm{g(x) = a(4)^x + 10}\)
  • Y-intercept is at point \(\mathrm{(0, 3)}\)

• What this tells us: When \(\mathrm{x = 0}\), the function value \(\mathrm{g(0) = 3}\)

2. INFER the solution approach

• To find the unknown parameter a, we need to use the y-intercept condition
• Since we know \(\mathrm{g(0) = 3}\), we can substitute \(\mathrm{x = 0}\) into our function and solve

3. SIMPLIFY by substituting x = 0

• Start with: \(\mathrm{g(0) = a(4)^0 + 10}\)
• Apply the exponential rule: \(\mathrm{4^0 = 1}\)
• This gives us: \(\mathrm{g(0) = a(1) + 10 = a + 10}\)

4. SIMPLIFY to solve for a

• Set up the equation: \(\mathrm{a + 10 = 3}\)
• Subtract 10 from both sides: \(\mathrm{a = 3 - 10}\)
• Final result: \(\mathrm{a = -7}\)

Answer: B) -7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not correctly interpret what "y-intercept at (0, 3)" means mathematically. They might think they need to substitute 3 for x, or they might not realize this gives them the condition \(\mathrm{g(0) = 3}\).

This confusion leads to setting up incorrect equations and typically results in guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{g(0) = 3}\) but make algebraic errors. The most common mistake is incorrectly evaluating \(\mathrm{4^0}\), either thinking it equals 0 or equals 4, rather than 1.

If they think \(\mathrm{4^0 = 0}\), they get: \(\mathrm{0 + 10 = 3}\), leading to \(\mathrm{10 = 3}\) (impossible)
If they think \(\mathrm{4^0 = 4}\), they get: \(\mathrm{4a + 10 = 3}\), leading to \(\mathrm{a = -7/4}\) (not among choices)

This may lead them to select Choice A (-11) or cause confusion and guessing.

The Bottom Line:

This problem tests whether students understand both the geometric meaning of y-intercept and the fundamental exponential property that any non-zero base to the power 0 equals 1. Success requires connecting the coordinate geometry concept to algebraic manipulation.