In the xy-plane, the graph of the linear function f contains the points \(\mathrm{(0, 3)}\) and \(\mathrm{(7, 31)}\). Which equation...

1. TRANSLATE the problem information

• Given information:
  • Point 1: \(\mathrm{(0, 3)}\)
  • Point 2: \(\mathrm{(7, 31)}\)
  • Need to find \(\mathrm{f(x)}\) in the form \(\mathrm{y = f(x)}\)
• What this tells us: We have two points on a line and need to find its equation

2. INFER the approach

• To write a linear equation, we need slope (m) and y-intercept (b) for the form \(\mathrm{f(x) = mx + b}\)
• The point \(\mathrm{(0, 3)}\) is special—it directly gives us the y-intercept since \(\mathrm{x = 0}\)
• We can use both points with the slope formula to find the slope

3. Identify the y-intercept

Since we have the point \(\mathrm{(0, 3)}\), the y-intercept is \(\mathrm{b = 3}\).

4. SIMPLIFY to find the slope

Using the slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)

\(\mathrm{m = \frac{31 - 3}{7 - 0}}\)
\(\mathrm{m = \frac{28}{7}}\)
\(\mathrm{m = 4}\)

5. Write the equation

\(\mathrm{f(x) = 4x + 3}\)

6. SIMPLIFY to verify with the second point

\(\mathrm{f(7) = 4(7) + 3}\)
\(\mathrm{f(7) = 28 + 3}\)
\(\mathrm{f(7) = 31}\) ✓

Answer: C. \(\mathrm{f(x) = 4x + 3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up the slope formula but make arithmetic errors, especially when calculating \(\mathrm{28 ÷ 7}\). Some might get confused and think \(\mathrm{\frac{28}{7} = 7}\) or miscalculate other ways.

This may lead them to select Choice D (\(\mathrm{f(x) = 7x + 3}\)) if they incorrectly think the slope is 7.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might mix up the coordinate order or not recognize that \(\mathrm{(0, 3)}\) immediately gives the y-intercept. Instead, they might try to use both points in a more complicated way or confuse x and y coordinates.

This causes confusion in their setup and may lead them to select any of the incorrect choices or abandon systematic solution and guess.

The Bottom Line:

This problem tests whether students can efficiently extract the key information from coordinate points—recognizing that \(\mathrm{(0, 3)}\) directly provides the y-intercept while both points together determine the slope. The arithmetic must be executed carefully to avoid selecting attractive wrong answers.