Function f is a quadratic function where \(\mathrm{f(-20) = 0}\) and \(\mathrm{f(-4) = 0}\). The graph of \(\mathrm{y = f(x)}\)...

1. TRANSLATE the problem information

• Given information:
  • f is a quadratic function
  • \(\mathrm{f(-20) = 0}\) and \(\mathrm{f(-4) = 0}\)
  • Vertex is at \(\mathrm{(r, -64)}\)
  • Need to find r
• What this tells us: Since \(\mathrm{f(-20) = 0}\) and \(\mathrm{f(-4) = 0}\), the points \(\mathrm{(-20, 0)}\) and \(\mathrm{(-4, 0)}\) are x-intercepts

2. INFER the key relationship

• For any quadratic function, the vertex lies exactly halfway between the x-intercepts
• This means the x-coordinate of the vertex equals the average of the x-intercept values
• Strategy: Find the midpoint of x = -20 and x = -4

3. SIMPLIFY to find the vertex x-coordinate

• x-coordinate of vertex = \(\frac{\mathrm{x_1 + x_2}}{2}\)
• x-coordinate of vertex = \(\frac{\mathrm{-20 + (-4)}}{2}\)
• x-coordinate of vertex = \(\frac{\mathrm{-24}}{2} = \mathrm{-12}\)

Since the vertex is at \(\mathrm{(r, -64)}\), we have \(\mathrm{r = -12}\).

Answer: -12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that \(\mathrm{f(-20) = 0}\) and \(\mathrm{f(-4) = 0}\) means these are x-intercepts. Instead, they might try to set up a system of equations or use the vertex form directly without understanding the relationship between zeros and x-intercepts. This leads to confusion and abandoning systematic solution, causing them to guess.

Second Most Common Error:

Missing conceptual knowledge about vertex location: Students might not know that the vertex x-coordinate is the average of the x-intercepts. They may attempt to use the vertex form \(\mathrm{f(x) = a(x - h)^2 + k}\) or try to complete the square, making the problem unnecessarily complex and leading to calculation errors or giving up.

The Bottom Line:

This problem tests whether students can connect the concept of function zeros to x-intercepts, and then apply the midpoint property of parabolas. The key insight is recognizing that \(\mathrm{f(x) = 0}\) gives you x-intercepts, and quadratic symmetry means the vertex lies exactly between them.