The function \(\mathrm{f(x) = -7(3)^{x+1} + 25}\). What is the y-intercept of the graph of \(\mathrm{y = f(x)}\) in the...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = -7(3)^{x+1} + 25}\)
  • Need to find: y-intercept of the graph
• What this tells us: The y-intercept occurs where the graph crosses the y-axis, which happens when \(\mathrm{x = 0}\)

2. INFER the approach

• To find the y-intercept, substitute \(\mathrm{x = 0}\) into the function
• This gives us the y-coordinate of the point where the graph crosses the y-axis

3. SIMPLIFY by substituting and evaluating

• Substitute \(\mathrm{x = 0}\): \(\mathrm{f(0) = -7(3)^{0+1} + 25}\)
• Simplify the exponent: \(\mathrm{f(0) = -7(3)^1 + 25}\)
• Evaluate \(\mathrm{3^1 = 3}\): \(\mathrm{f(0) = -7(3) + 25}\)
• Calculate \(\mathrm{-7(3) = -21}\): \(\mathrm{f(0) = -21 + 25}\)
• Final calculation: \(\mathrm{f(0) = 4}\)

4. INFER the coordinate form

• The y-intercept is the point \(\mathrm{(0, y\text{-}value)}\)
• Since \(\mathrm{f(0) = 4}\), the y-intercept is \(\mathrm{(0, 4)}\)

Answer: A. \(\mathrm{(0, 4)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students may not understand that "y-intercept" specifically means finding the function value when \(\mathrm{x = 0}\).

Some students might try to set \(\mathrm{f(x) = 0}\) to find where the graph crosses the x-axis instead, leading them to solve \(\mathrm{-7(3)^{x+1} + 25 = 0}\). This creates unnecessary complexity and confusion, causing them to get stuck and guess randomly among the answer choices.

Second Most Common Error:

Weak SIMPLIFY execution: Students make arithmetic errors during the calculation process.

Common computational mistakes include:

• Calculating \(\mathrm{3^1}\) incorrectly (though this is rare)
• Getting \(\mathrm{-7(3) = 21}\) instead of \(\mathrm{-21}\) (sign error)
• Adding \(\mathrm{-21 + 25}\) incorrectly to get 46 or -46

These calculation errors might lead them to select Choice B \(\mathrm{(0, 18)}\) if they get \(\mathrm{-7 + 25 = 18}\) by forgetting to multiply by 3, or other incorrect choices.

The Bottom Line:

This problem tests whether students understand the fundamental definition of y-intercept and can execute straightforward function evaluation without computational errors.