A linear function is defined by \(\mathrm{f(x) = ax + b}\), where a and b are constants. The graph of...

1. TRANSLATE the problem information

• Given information:
  • Linear function: \(\mathrm{f(x) = ax + b}\)
  • Passes through points \(\mathrm{(-2, -7)}\) and \(\mathrm{(4, 5)}\)
  • Need to find: \(\mathrm{a + b}\)

2. INFER the solution strategy

• To find \(\mathrm{a + b}\), we need to find both \(\mathrm{a}\) (slope) and \(\mathrm{b}\) (y-intercept) separately
• With two points, we can use the slope formula to find \(\mathrm{a}\)
• Once we have \(\mathrm{a}\), we can substitute either point into \(\mathrm{f(x) = ax + b}\) to find \(\mathrm{b}\)

3. SIMPLIFY to find the slope a

• Apply slope formula: \(\mathrm{a = \frac{y_2 - y_1}{x_2 - x_1}}\)
• Using \(\mathrm{(-2, -7)}\) as \(\mathrm{(x_1, y_1)}\) and \(\mathrm{(4, 5)}\) as \(\mathrm{(x_2, y_2)}\):
  \(\mathrm{a = \frac{5 - (-7)}{4 - (-2)}}\)
  \(\mathrm{= \frac{5 + 7}{6}}\)
  \(\mathrm{= \frac{12}{6}}\)
  \(\mathrm{= 2}\)

4. SIMPLIFY to find the y-intercept b

• Substitute \(\mathrm{a = 2}\) and point \(\mathrm{(4, 5)}\) into \(\mathrm{f(x) = ax + b}\):
  \(\mathrm{5 = 2(4) + b}\)
  \(\mathrm{5 = 8 + b}\)
  \(\mathrm{b = 5 - 8 = -3}\)

5. Verify with the other point (optional but recommended)

• Check: \(\mathrm{f(-2) = 2(-2) + (-3)}\)
  \(\mathrm{= -4 - 3}\)
  \(\mathrm{= -7}\) ✓

6. SIMPLIFY to find the final answer

• \(\mathrm{a + b = 2 + (-3) = -1}\)

Answer: B. -1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making sign errors when calculating the slope

Students often mess up the subtraction in the slope formula, especially with negative coordinates. For example, calculating \(\mathrm{(5 - (-7))}\) as \(\mathrm{(5 - 7) = -2}\) instead of \(\mathrm{(5 + 7) = 12}\), leading to \(\mathrm{a = \frac{-2}{6} = -\frac{1}{3}}\). Then finding \(\mathrm{b}\) incorrectly and getting a wrong sum.

This may lead them to select Choice A (-5) or cause confusion leading to guessing.

Second Most Common Error:

Inadequate INFER reasoning: Mixing up which parameter is which

Students sometimes confuse the slope and y-intercept, or forget that they need both values to answer the question. They might find the slope correctly but then assume that's the final answer, or they might try to find the y-intercept directly without first finding the slope.

This leads to confusion and guessing among the choices.

The Bottom Line:

This problem tests whether students can systematically work through finding both parameters of a linear function from two points. Success requires careful arithmetic with signed numbers and staying organized through the multi-step process.