xy1112193aThe table shows three values of x and their corresponding values of y for the equation \(\mathrm{y = 4(2)^x +...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(\mathrm{y = 4(2)^x + 3}\)
  • Table shows that when \(\mathrm{x = 3}\), the corresponding y-value is a
• What we need to find: The value of constant a

2. INFER the solution approach

• Since we know the equation and the x-value (\(\mathrm{x = 3}\)), we can substitute directly
• The equation will give us the y-value, which equals a

3. SIMPLIFY by substituting and evaluating

• Substitute \(\mathrm{x = 3}\) into \(\mathrm{y = 4(2)^x + 3}\):
  • \(\mathrm{a = 4(2)^3 + 3}\)
• Evaluate the exponent first: \(\mathrm{2^3 = 8}\)
  • \(\mathrm{a = 4(8) + 3}\)
• Multiply: \(\mathrm{4 \times 8 = 32}\)
  • \(\mathrm{a = 32 + 3}\)
• Add: \(\mathrm{a = 35}\)

Answer: B. 35

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make order of operations errors when evaluating \(\mathrm{4(2)^3 + 3}\)

Many students incorrectly calculate \(\mathrm{(4\times2)^3}\) instead of \(\mathrm{4\times(2^3)}\), getting:

• \(\mathrm{(4\times2)^3 + 3 = 8^3 + 3 = 512 + 3 = 515}\)

Since 515 isn't among the choices, this leads to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might try to use the other table values instead of directly substituting

They might attempt to find a pattern from \(\mathrm{x=1\to y=11}\) and \(\mathrm{x=2\to y=19}\), then try to extend it to \(\mathrm{x=3}\). This approach is unnecessarily complicated and prone to arithmetic errors, potentially leading them to select Choice D (27) if they incorrectly identify the pattern as "add 8 each time."

The Bottom Line:

This problem tests whether students can correctly substitute into an exponential function and follow order of operations. The key insight is recognizing that you can directly use the given equation rather than trying to find patterns in the table values.