For the linear function g, the graph of \(\mathrm{y = g(x)}\) in the xy-plane has a slope of -5. The...

1. TRANSLATE the problem information

• Given information:
  • Slope = -5
  • x-intercept at point (2, 0)
• This means we need to find a linear equation where the line has slope -5 and crosses the x-axis at x = 2

2. INFER the solution approach

• Since we know the slope, we can start with slope-intercept form: \(\mathrm{g(x) = mx + b}\)
• We have \(\mathrm{m = -5}\), so \(\mathrm{g(x) = -5x + b}\)
• We need to find the y-intercept (b) using the x-intercept information

3. TRANSLATE the x-intercept into a coordinate point

• x-intercept at \(\mathrm{(2, 0)}\) means: when \(\mathrm{x = 2}\), the function value \(\mathrm{g(2) = 0}\)
• This gives us a point we can substitute into our equation

4. SIMPLIFY to find the y-intercept

• Substitute the point \(\mathrm{(2, 0)}\) into \(\mathrm{g(x) = -5x + b}\):
  \(\mathrm{0 = -5(2) + b}\)
  \(\mathrm{0 = -10 + b}\)
  \(\mathrm{b = 10}\)

5. Write the final equation

• Now we have both \(\mathrm{m = -5}\) and \(\mathrm{b = 10}\)
• Therefore: \(\mathrm{g(x) = -5x + 10}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse x-intercept with y-intercept, thinking the point \(\mathrm{(2, 0)}\) gives them the y-intercept directly as \(\mathrm{b = 0}\).

They might write \(\mathrm{g(x) = -5x + 0 = -5x}\), but this isn't among the choices, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when solving \(\mathrm{0 = -10 + b}\), getting \(\mathrm{b = -10}\) instead of \(\mathrm{b = 10}\).

This leads them to select Choice A (\(\mathrm{g(x) = -5x - 10}\)).

The Bottom Line:

This problem requires students to understand that an x-intercept is a point on the line, not just a number, and that this point can be used to find the missing parameter in the linear equation.