In the xy-plane, the graph of the linear function f passes through the point \((9, -2)\). The graph of f...

1. TRANSLATE the problem information

• Given information:
  • Function f passes through point (9, -2)
  • X-intercept at -3

• What this tells us: An x-intercept at -3 means the function passes through point (-3, 0)

2. INFER the approach

• With two points on a linear function, we can find the slope and then the equation
• Strategy: Calculate slope → Use point-slope form → Convert to slope-intercept form

3. SIMPLIFY to find the slope

• Using points (9, -2) and (-3, 0):

\(\mathrm{m = \frac{0 - (-2)}{-3 - 9}}\)

\(\mathrm{= \frac{2}{-12}}\)

\(\mathrm{= -\frac{1}{6}}\)

4. APPLY point-slope form

• Using point (-3, 0) and slope m = -1/6:

\(\mathrm{y - 0 = -\frac{1}{6}(x - (-3))}\)

\(\mathrm{y = -\frac{1}{6}(x + 3)}\)

5. SIMPLIFY to slope-intercept form

• Distribute:

\(\mathrm{y = -\frac{1}{6}x + (-\frac{1}{6})(3)}\)

• Calculate:

\(\mathrm{y = -\frac{1}{6}x - \frac{1}{2}}\)

Answer: C. \(\mathrm{f(x) = -\frac{1}{6}x - \frac{1}{2}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students miss that "x-intercept at -3" means the point (-3, 0). They might try to work with incomplete information or incorrectly assume the intercept gives them a y-value.

Without the second coordinate point, they can't calculate slope and either get stuck or try to guess based on the given point alone. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly find the slope but make algebraic errors when distributing -1/6 across (x + 3). Common mistakes include:

• Getting \(\mathrm{-\frac{1}{6}x + 3}\) instead of \(\mathrm{-\frac{1}{6}x - \frac{1}{2}}\)
• Sign errors with the constant term
• Fraction arithmetic mistakes

This may lead them to select Choice A (-1/3x + 1) or Choice D (1/6x + 1/2) depending on their specific error.

The Bottom Line:

Success requires recognizing that intercepts are coordinate points, not just single numbers. The x-intercept at -3 gives you the complete point (-3, 0), which becomes your second piece of information for finding the linear equation.