The function f is defined by \(\mathrm{f(x) = a^x + b}\), where a and b are constants and a gt...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = a^x + b}\) where \(\mathrm{a \gt 0}\)
  • Y-intercept at \(\mathrm{(0, -25)}\), meaning \(\mathrm{f(0) = -25}\)
  • Point \(\mathrm{(2, 23)}\) is on the graph, meaning \(\mathrm{f(2) = 23}\)
  • Need to find: \(\mathrm{a + b}\)

2. INFER the solution approach

• Use the y-intercept first since \(\mathrm{a^0 = 1}\) makes this straightforward
• Then use the second point to find a
• Apply the constraint \(\mathrm{a \gt 0}\) if needed

3. SIMPLIFY using the y-intercept

• Substitute \(\mathrm{x = 0}\): \(\mathrm{f(0) = a^0 + b = 1 + b = -25}\)
• Solve: \(\mathrm{b = -26}\)

4. SIMPLIFY using the second point

• Substitute \(\mathrm{x = 2}\): \(\mathrm{f(2) = a^2 + b = a^2 + (-26) = 23}\)
• Add 26 to both sides: \(\mathrm{a^2 = 49}\)
• Take square root: \(\mathrm{a = \pm7}\)

5. APPLY CONSTRAINTS to select final answer

• Since \(\mathrm{a \gt 0}\): \(\mathrm{a = 7}\)
• Therefore: \(\mathrm{a + b = 7 + (-26) = -19}\)

Answer: -19

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \(\mathrm{a^0 = 1}\), so they get stuck trying to work with the y-intercept condition. Instead, they might try to use both points simultaneously in a system of equations, making the algebra much more complex and error-prone. This leads to confusion and potentially abandoning the systematic approach.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students find \(\mathrm{a^2 = 49}\) and get \(\mathrm{a = \pm7}\), but forget to apply the given constraint \(\mathrm{a \gt 0}\). They might arbitrarily choose \(\mathrm{a = -7}\), leading to \(\mathrm{a + b = -7 + (-26) = -33}\). This incorrect reasoning would lead them to select a wrong answer if -33 were among the choices, or cause confusion when their answer doesn't match any option.

The Bottom Line:

This problem tests whether students can efficiently use the special property of exponential functions at \(\mathrm{x = 0}\), combined with careful attention to given constraints. The key insight is recognizing that \(\mathrm{a^0 = 1}\) makes the y-intercept condition immediately solvable for one parameter.