The graph of \(\mathrm{y = f(x - 5)}\) is shown above. Which of the following equations defines function f?

1. TRANSLATE the graph into an equation

First, we need to find the equation of the line shown in the graph.

• Identify two clear points on the line:
  • x-intercept: \((-2, 0)\)
  • y-intercept: \((0, 4)\)

• Calculate the slope:
  • \(\mathrm{m} = \frac{\mathrm{y_2 - y_1}}{\mathrm{x_2 - x_1}} = \frac{4 - 0}{0 - (-2)} = \frac{4}{2} = 2\)

• Identify the y-intercept:
  • \(\mathrm{b} = 4\) (where the line crosses the y-axis)

• Write the equation:
  • The line shown is \(\mathrm{y} = 2\mathrm{x} + 4\)

2. TRANSLATE the problem statement

The problem tells us "the graph of \(\mathrm{y} = \mathrm{f(x - 5)}\) is shown."

This means: \(\mathrm{f(x - 5)} = 2\mathrm{x} + 4\)

This is crucial! The graph doesn't show \(\mathrm{f(x)}\) directly—it shows what happens when we input \(\mathrm{x} - 5\) into function f.

3. INFER the strategy to find f(x)

We have \(\mathrm{f(x - 5)} = 2\mathrm{x} + 4\), but we need \(\mathrm{f(x)}\).

Key strategic insight: Use substitution to "undo" the transformation.

• Let \(\mathrm{u} = \mathrm{x} - 5\)
• This means \(\mathrm{x} = \mathrm{u} + 5\)
• Wherever we see x in our equation, replace it with \(\mathrm{u} + 5\)

4. SIMPLIFY using substitution

Starting with: \(\mathrm{f(x - 5)} = 2\mathrm{x} + 4\)

Since \(\mathrm{u} = \mathrm{x} - 5\), we have \(\mathrm{x} = \mathrm{u} + 5\), so:

\(\mathrm{f(u)} = 2(\mathrm{u} + 5) + 4\)

Expand:

• \(\mathrm{f(u)} = 2\mathrm{u} + 10 + 4\)
• \(\mathrm{f(u)} = 2\mathrm{u} + 14\)

Since u is just a placeholder variable, we can write:

\(\mathrm{f(x)} = 2\mathrm{x} + 14\)

5. Match with answer choices

Looking at our options, \(\mathrm{f(x)} = 2\mathrm{x} + 14\) is choice C.

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Process Skill Gap - Weak TRANSLATE: Students may misinterpret what the problem is asking. They see the graph shows \(\mathrm{y} = 2\mathrm{x} + 4\) and think "that must be \(\mathrm{f(x)}\)!" They don't properly translate the statement "the graph of \(\mathrm{y} = \mathrm{f(x - 5)}\)" into the mathematical relationship \(\mathrm{f(x - 5)} = 2\mathrm{x} + 4\).

Without recognizing that the graph shows \(\mathrm{f(x - 5)}\), not \(\mathrm{f(x)}\), they immediately select Choice B (\(\mathrm{f(x)} = 2\mathrm{x} + 4\)) because it matches what they see on the graph.

This is the trap answer—it looks right if you miss the transformation in the problem statement.

Second Most Common Error:

Process Skill Gap - Weak INFER/SIMPLIFY: Students correctly identify that \(\mathrm{f(x - 5)} = 2\mathrm{x} + 4\), but then make errors in the substitution or algebraic manipulation.

Common mistakes include:

• Setting \(\mathrm{f(x)} = 2(\mathrm{x} - 5) + 4\), incorrectly thinking they should substitute \(\mathrm{x} - 5\) for x in the original equation, leading to \(\mathrm{f(x)} = 2\mathrm{x} - 10 + 4 = 2\mathrm{x} - 6\), which is Choice A
• Making arithmetic errors when expanding \(2(\mathrm{u} + 5) + 4\), possibly getting \(2\mathrm{u} + 6\) instead of \(2\mathrm{u} + 14\)

These algebraic errors cause students to select incorrect answer choices even though they understood the conceptual approach.

The Bottom Line:

This problem tests whether students can distinguish between a function and its transformation. The critical moment is recognizing that seeing "\(\mathrm{y} = \mathrm{f(x - 5)}\)" means the graph doesn't show the original function—it shows a modified version. Once that translation is clear, the rest is systematic algebra.