For the linear function g, the graph of \(\mathrm{y = g(x)}\) in the xy-plane passes through the points \(\mathrm{(2, 7)}\)...

1. TRANSLATE the problem information

• Given information:
  • Linear function g passes through \(\mathrm{(2, 7)}\) and \(\mathrm{(6, 9)}\)
  • Need to find the equation that defines g
• What this tells us: We have two points on the line, which is enough information to determine a unique linear function

2. INFER the solution approach

• To define a linear function, we need both the slope (m) and y-intercept (b) for the form \(\mathrm{y = mx + b}\)
• Strategy: First calculate slope, then use point-slope form to find the complete equation

3. SIMPLIFY to find the slope

• Using the slope formula with points \(\mathrm{(2, 7)}\) and \(\mathrm{(6, 9)}\):

\(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)

\(\mathrm{m = \frac{9 - 7}{6 - 2}}\)

\(\mathrm{m = \frac{2}{4}}\)

\(\mathrm{m = \frac{1}{2}}\)

4. SIMPLIFY to find the complete equation

• Using point-slope form with slope \(\mathrm{\frac{1}{2}}\) and point \(\mathrm{(2, 7)}\):

\(\mathrm{y - 7 = \frac{1}{2}(x - 2)}\)

\(\mathrm{y - 7 = \frac{1}{2}x - 1}\)

\(\mathrm{y = \frac{1}{2}x - 1 + 7}\)

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

5. Verify the solution

• Check with the other point \(\mathrm{(6, 9)}\): \(\mathrm{g(6) = \frac{1}{2}(6) + 6 = 3 + 6 = 9}\) ✓

Answer: B) \(\mathrm{g(x) = \frac{1}{2}x + 6}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making an arithmetic error when calculating the slope fraction

Students might incorrectly simplify \(\mathrm{\frac{2}{4}}\), getting 2 instead of \(\mathrm{\frac{1}{2}}\). This leads them to work with slope \(\mathrm{m = 2}\), and when they apply point-slope form:

\(\mathrm{y - 7 = 2(x - 2)}\)

\(\mathrm{y = 2x - 4 + 7}\)

\(\mathrm{y = 2x + 3}\)

This may lead them to select Choice A (\(\mathrm{g(x) = 2x + 3}\))

Second Most Common Error:

Poor SIMPLIFY execution: Correct slope but algebraic mistakes in point-slope manipulation

Students get the slope right \(\mathrm{(\frac{1}{2})}\) but make sign errors or arithmetic mistakes when distributing or combining terms in the point-slope form. For example, incorrectly getting \(\mathrm{y = \frac{1}{2}x + 7}\) or \(\mathrm{y = \frac{1}{2}x + 8}\) instead of \(\mathrm{y = \frac{1}{2}x + 6}\).

This may lead them to select Choice C (\(\mathrm{g(x) = \frac{1}{2}x + 7}\)) or Choice D (\(\mathrm{g(x) = \frac{1}{2}x + 8}\))

The Bottom Line:

This problem tests both computational accuracy and algebraic manipulation skills. The slope calculation seems straightforward but requires careful fraction arithmetic, while the point-slope algebra demands systematic step-by-step work to avoid sign and arithmetic errors.