In the xy-plane, the graph of the linear function f contains the points \(\mathrm{(0, 2)}\) and \(\mathrm{(8, 34)}\). Which equation...

1. TRANSLATE the problem information

• Given information:
  • Linear function f contains points \(\mathrm{(0, 2)}\) and \(\mathrm{(8, 34)}\)
  • Need to find equation \(\mathrm{y = f(x)}\)

2. INFER the strategic approach

• Since this is a linear function, it has the form \(\mathrm{f(x) = mx + b}\)
• Key insight: Point \(\mathrm{(0, 2)}\) means when \(\mathrm{x = 0, y = 2}\), so the y-intercept \(\mathrm{b = 2}\)
• Need to find slope m using the slope formula with both given points

3. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• Substituting points \(\mathrm{(0, 2)}\) and \(\mathrm{(8, 34)}\):
  \(\mathrm{m = \frac{34 - 2}{8 - 0}}\)
  \(\mathrm{m = \frac{32}{8}}\)
  \(\mathrm{m = 4}\)

4. Combine the components

• With \(\mathrm{m = 4}\) and \(\mathrm{b = 2}\): \(\mathrm{f(x) = 4x + 2}\)

Answer: C. f(x) = 4x + 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students correctly set up the slope calculation as \(\mathrm{\frac{34-2}{8-0} = \frac{32}{8}}\), but then use 32 as the slope without dividing by 8.

They think: "The numerator is 32, so the slope is 32." This leads to an equation like \(\mathrm{f(x) = 32x + \text{something}}\), and they might select Choice B (\(\mathrm{f(x) = 32x + 36}\)).

Second Most Common Error:

Poor INFER reasoning: Students get confused about which number represents the slope. They might use the x-coordinate 8 from point \(\mathrm{(8, 34)}\) as the slope, thinking "8 looks like a reasonable slope value."

This leads them to create \(\mathrm{f(x) = 8x + 2}\) and select Choice D (\(\mathrm{f(x) = 8x + 2}\)).

The Bottom Line:

Success requires recognizing that \(\mathrm{(0, 2)}\) immediately gives you the y-intercept, then carefully calculating and simplifying the slope fraction. The arithmetic step of \(\mathrm{32 \div 8 = 4}\) is crucial - skipping this division is the most common mistake.