In the linear function f, \(\mathrm{f(0) = 8}\) and \(\mathrm{f(1) = 12}\). Which equation defines f?

1. TRANSLATE the problem information

• Given information:
  • f is a linear function
  • \(\mathrm{f(0) = 8}\)
  • \(\mathrm{f(1) = 12}\)
• We need to find which equation defines f

2. INFER the approach

• Since f is linear, it has the form \(\mathrm{f(x) = ax + b}\) where a is the slope and b is the y-intercept
• The key insight: \(\mathrm{f(0)}\) will directly give us the y-intercept since \(\mathrm{f(0) = a(0) + b = b}\)
• Then \(\mathrm{f(1)}\) will help us find the slope

3. Find the y-intercept using \(\mathrm{f(0) = 8}\)

• Substitute into \(\mathrm{f(x) = ax + b}\):

\(\mathrm{f(0) = a(0) + b = b = 8}\)

• So \(\mathrm{b = 8}\), giving us \(\mathrm{f(x) = ax + 8}\)

4. SIMPLIFY to find the slope using \(\mathrm{f(1) = 12}\)

• Substitute into \(\mathrm{f(x) = ax + 8}\):

\(\mathrm{f(1) = a(1) + 8 = a + 8 = 12}\)

• Solve: \(\mathrm{a = 12 - 8 = 4}\)

5. Write the complete function

• \(\mathrm{f(x) = 4x + 8}\)

Answer: D. \(\mathrm{f(x) = 4x + 8}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students might misinterpret what \(\mathrm{f(1) = 12}\) means and think the slope should be 12 instead of recognizing that the slope is the change from \(\mathrm{f(0)}\) to \(\mathrm{f(1)}\).

They might reason: "\(\mathrm{f(1) = 12}\), so the coefficient of x should be 12" and incorrectly conclude \(\mathrm{f(x) = 12x + 8}\).

This may lead them to select Choice A (\(\mathrm{f(x) = 12x + 8}\)).

Second Most Common Error:

Conceptual confusion about function components: Students might confuse which value corresponds to which parameter in \(\mathrm{f(x) = ax + b}\), thinking \(\mathrm{f(1) = 12}\) means the y-intercept is 12 rather than using the systematic approach.

They might incorrectly write \(\mathrm{f(x) = 4x + 12}\), getting the slope right but misplacing the y-intercept.

This may lead them to select Choice C (\(\mathrm{f(x) = 4x + 12}\)).

The Bottom Line:

Success requires recognizing that \(\mathrm{f(0)}\) immediately gives the y-intercept in the slope-intercept form, then using that information strategically with the second condition to find the slope. Students who try to work backward from answer choices or who don't systematically use the function form often make parameter placement errors.