x\(\mathrm{p(x)}\)-2302-2516The table shows three values of x and their corresponding values of \(\mathrm{p(x)}\), where \(\mathrm{p(x) = f(x)(x-1)}\)...

1. INFER the key relationship

• Given: \(\mathrm{p(x) = f(x)(x-1)}\) where f is linear
• Key insight: We can find specific values of \(\mathrm{f(x)}\) by rearranging this equation
• Since \(\mathrm{p(x) = f(x)(x-1)}\), then \(\mathrm{f(x) = \frac{p(x)}{x-1}}\)

2. SIMPLIFY to find f(x) values

• Calculate f(x) for each given point:
  • \(\mathrm{f(-2) = \frac{30}{-3} = -10}\)
  • \(\mathrm{f(2) = \frac{-2}{1} = -2}\)
  • \(\mathrm{f(5) = \frac{16}{4} = 4}\)
• Now we have three points on the linear function f(x): (-2, -10), (2, -2), and (5, 4)

3. INFER that we need the slope first

• Since f(x) is linear, it has form \(\mathrm{f(x) = mx + b}\)
• To find b (which gives us the y-intercept), we first need the slope m

4. SIMPLIFY the slope calculation

• Using points (2, -2) and (5, 4):
• \(\mathrm{m = \frac{4-(-2)}{5-2} = \frac{6}{3} = 2}\)

5. SIMPLIFY to find the y-intercept

• Use point-slope form with point (2, -2):
• \(\mathrm{-2 = 2(2) + b}\)
• \(\mathrm{-2 = 4 + b}\)
• \(\mathrm{b = -6}\)
• Therefore \(\mathrm{f(x) = 2x - 6}\), and the y-intercept is (0, -6)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they can find f(x) from the given relationship \(\mathrm{p(x) = f(x)(x-1)}\). Instead, they try to work directly with p(x) values or attempt to guess the form of f(x) without using the constraint that \(\mathrm{p(x) = f(x)(x-1)}\). This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{f(x) = \frac{p(x)}{x-1}}\) but make arithmetic errors when calculating the divisions, particularly \(\mathrm{f(-2) = \frac{30}{-3} = -10}\). Getting f(-2) = 10 instead of -10 changes the slope calculation completely. This may lead them to select Choice D (0, 4).

The Bottom Line:

This problem requires students to see that a constraint equation can be rearranged to extract the information they need. The key breakthrough is realizing that \(\mathrm{p(x) = f(x)(x-1)}\) isn't just given information - it's a tool for finding f(x) values.