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

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = a^x + b}\) (exponential function with constants a and b)
  • y-intercept at \(\mathrm{(0, -323)}\)
  • x-intercept at \(\mathrm{(2, 0)}\)
• What this tells us: The y-intercept means \(\mathrm{f(0) = -323}\)

2. INFER which intercept to use

• We need to find b, so we should use the y-intercept condition
• The y-intercept gives us \(\mathrm{f(0) = -323}\), which will help us find b directly
• The x-intercept would give us \(\mathrm{f(2) = 0}\), but this involves both a and b

3. TRANSLATE the y-intercept condition into an equation

• \(\mathrm{f(0) = -323}\)
• Substituting into \(\mathrm{f(x) = a^x + b}\):
• \(\mathrm{a^0 + b = -323}\)

4. INFER the value of a⁰

• Any non-zero number raised to the power 0 equals 1
• Therefore: \(\mathrm{a^0 = 1}\)
• Our equation becomes: \(\mathrm{1 + b = -323}\)

5. SIMPLIFY to solve for b

• \(\mathrm{1 + b = -323}\)
• Subtract 1 from both sides: \(\mathrm{b = -323 - 1}\)
• \(\mathrm{b = -324}\)

Answer: -324

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might try to use the x-intercept instead of the y-intercept to find b.

They substitute \(\mathrm{f(2) = 0}\) into the equation: \(\mathrm{a^2 + b = 0}\), giving \(\mathrm{b = -a^2}\). Since they don't know the value of a, they get stuck or try to use both intercepts simultaneously, creating unnecessary complexity. This leads to confusion and guessing.

Second Most Common Error:

Missing conceptual knowledge about exponents: Students forget that \(\mathrm{a^0 = 1}\).

They see \(\mathrm{a^0 + b = -323}\) and either treat \(\mathrm{a^0}\) as 0 (getting \(\mathrm{b = -323}\)) or leave it as an unknown variable. This may lead them to think they need more information to solve the problem, causing them to abandon the systematic approach and guess.

The Bottom Line:

This problem tests whether students can efficiently choose which given information to use first and whether they remember the fundamental exponent property \(\mathrm{a^0 = 1}\). The key insight is recognizing that the y-intercept gives direct access to finding b.