Question:A quadratic function f models the height (in feet) of a projectile over time, where f has x-intercepts at x...

1. TRANSLATE the problem information into coordinate points

• Given information about function f:
  • "x-intercepts at x = -2 and x = 5" → \(\mathrm{f(-2) = 0}\) and \(\mathrm{f(5) = 0}\)
  • "minimum value of -12 at x = 1.5" → \(\mathrm{f(1.5) = -12}\)

• What this tells us: We know three specific points on function f: \(\mathrm{(-2, 0)}\), \(\mathrm{(1.5, -12)}\), and \(\mathrm{(5, 0)}\)

2. INFER how the transformation affects these points

• The transformation \(\mathrm{g(x) = 2.5f(x)}\) is a vertical scaling
• Key insight: Vertical scaling multiplies every y-coordinate by the scale factor, but x-coordinates stay the same
• Strategy: Calculate \(\mathrm{g(x)}\) for each of our known x-values using \(\mathrm{g(x) = 2.5f(x)}\)

3. SIMPLIFY to find the new y-coordinates

• For \(\mathrm{x = -2}\): \(\mathrm{g(-2) = 2.5 \times f(-2)}\)
  \(\mathrm{= 2.5 \times 0}\)
  \(\mathrm{= 0}\)
• For \(\mathrm{x = 1.5}\): \(\mathrm{g(1.5) = 2.5 \times f(1.5)}\)
  \(\mathrm{= 2.5 \times (-12)}\)
  \(\mathrm{= -30}\)
• For \(\mathrm{x = 5}\): \(\mathrm{g(5) = 2.5 \times f(5)}\)
  \(\mathrm{= 2.5 \times 0}\)
  \(\mathrm{= 0}\)

4. Match the transformed points to the answer choices

• The three points for function g are: \(\mathrm{(-2, 0)}\), \(\mathrm{(1.5, -30)}\), and \(\mathrm{(5, 0)}\)
• Only Choice A contains all three of these coordinate pairs

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may think the transformation \(\mathrm{g(x) = 2.5f(x)}\) scales both x and y coordinates, leading them to calculate new x-values as well as new y-values.

For example, they might think the x-intercepts become \(\mathrm{x = 2.5(-2) = -5}\) and \(\mathrm{x = 2.5(5) = 12.5}\), while keeping some y-values unchanged. This incorrect reasoning about how vertical scaling works may lead them to select Choice B which has x-intercepts at -5 and 12.5.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly understand the transformation but make an arithmetic error when calculating \(\mathrm{2.5 \times (-12)}\).

If they incorrectly compute \(\mathrm{2.5 \times (-12) = -9.5}\) instead of -30, they might select Choice C which has the point \(\mathrm{(1.5, -9.5)}\).

The Bottom Line:

This problem tests whether students understand that vertical scaling affects only y-coordinates, not x-coordinates. The key insight is recognizing that \(\mathrm{g(x) = 2.5f(x)}\) means "take whatever f outputs and multiply it by 2.5" - the input values (x-coordinates) remain exactly the same.