The table above shows some values of x and their corresponding values \(\mathrm{f(x)}\) for the linear function f. What is...

1. TRANSLATE the problem information

• Given information:
  • Table showing x-values and corresponding f(x) values for a linear function
  • Need to find: x-intercept of \(\mathrm{y = f(x)}\)
• What this tells us: We have coordinate points \(\mathrm{(-11, 21)}\), \(\mathrm{(-10, 18)}\), \(\mathrm{(-9, 15)}\), \(\mathrm{(-8, 12)}\), and we need the point where \(\mathrm{y = 0}\)

2. INFER the approach

• Since we have a linear function, we can write it as \(\mathrm{y = mx + b}\)
• Strategy: Find the equation first, then set \(\mathrm{y = 0}\) to find where the line crosses the x-axis
• We need slope (m) and y-intercept (b) to get the complete equation

3. SIMPLIFY to find the slope

• Using slope formula with points \(\mathrm{(-11, 21)}\) and \(\mathrm{(-10, 18)}\):
• \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• \(\mathrm{m = \frac{18 - 21}{-10 - (-11)}}\)
• \(\mathrm{m = \frac{-3}{1}}\)
• \(\mathrm{m = -3}\)

4. SIMPLIFY to find the y-intercept

• Substitute \(\mathrm{m = -3}\) and point \(\mathrm{(-11, 21)}\) into \(\mathrm{y = mx + b}\):
• \(\mathrm{21 = -3(-11) + b}\)
• \(\mathrm{21 = 33 + b}\)
• \(\mathrm{b = -12}\)

5. INFER the complete equation and solve for x-intercept

• Our equation is \(\mathrm{y = -3x - 12}\)
• For x-intercept, set \(\mathrm{y = 0}\):
• \(\mathrm{0 = -3x - 12}\)
• SIMPLIFY:
• \(\mathrm{-3x = 12}\)
• \(\mathrm{x = -4}\)
• The x-intercept is \(\mathrm{(-4, 0)}\)

Answer: B. (-4,0)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may try to find patterns in the table without recognizing they need the complete linear equation first. They might attempt to extend the pattern by continuing the sequence (noticing f(x) decreases by 3 each time) but get confused about where exactly \(\mathrm{y = 0}\) occurs.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the need to find the linear equation but make calculation errors. Common mistakes include:

• Getting the wrong slope due to sign errors in the slope formula
• Making arithmetic mistakes when solving for b
• Algebraic errors when solving \(\mathrm{0 = -3x - 12}\)

These calculation errors can lead them to select Choice A (-3,0) (confusing the slope with the x-coordinate) or Choice D (-12,0) (confusing the y-intercept with the x-coordinate).

The Bottom Line:

This problem tests whether students understand that finding intercepts requires the complete equation first, rather than trying to work directly from table patterns. The key insight is recognizing that systematic equation-building leads to a reliable solution method.