The graph of \(\mathrm{y = f(x) - 11}\) is shown. Which equation defines the linear function f?

1. TRANSLATE the graph information

The key insight here is recognizing what you're looking at: The graph shows y = f(x) - 11, NOT the graph of f(x) itself.

Your first task: Find the equation of the line that IS shown.

• TRANSLATE coordinates from the graph:
  • Point 1: (-2, 0)
  • Point 2: (-1, -2)
  • Point 3: (0, -4)

2. Calculate the slope of the shown line

Using any two points, apply the slope formula. Let's use (-1, -2) and (0, -4):

\(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
\(\mathrm{m = \frac{-4 - (-2)}{0 - (-1)}}\)
\(\mathrm{m = \frac{-2}{1}}\)
\(\mathrm{m = -2}\)

3. Identify the y-intercept

The line crosses the y-axis at the point (0, -4).

Therefore, b = -4

4. Write the equation of the shown line

Using \(\mathrm{y = mx + b}\):
\(\mathrm{y = -2x + (-4)}\)
\(\mathrm{y = -2x - 4}\)

This is the equation of the line you see in the graph.

5. INFER the relationship to find f(x)

Here's the crucial strategic thinking: The graph shows \(\mathrm{y = f(x) - 11}\), and we just found that this graph has equation \(\mathrm{y = -2x - 4}\).

Therefore:
\(\mathrm{f(x) - 11 = -2x - 4}\)

6. SIMPLIFY to solve for f(x)

\(\mathrm{f(x) - 11 = -2x - 4}\)

Add 11 to both sides:
\(\mathrm{f(x) = -2x - 4 + 11}\)
\(\mathrm{f(x) = -2x + 7}\)

Answer: B. f(x) = -2x + 7

Verification Check:

If \(\mathrm{f(x) = -2x + 7}\), then \(\mathrm{f(x) - 11 = -2x + 7 - 11 = -2x - 4}\) ✓
This matches our line!

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Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misunderstanding what the graph represents

Students often think the graph shows f(x) directly, forgetting that it actually shows f(x) - 11. They find the equation y = -2x - 4 from the graph and then incorrectly conclude that f(x) = -2x - 4.

This leads them to select Choice D (f(x) = -2x - 11) when they try to "match" their equation with answer choices by seeing the -11 and thinking it relates to the graph transformation, or it causes confusion and guessing among the choices.

Second Most Common Error:

Weak SIMPLIFY execution: Sign errors when solving for f(x)

Students correctly set up f(x) - 11 = -2x - 4, but make arithmetic errors when adding 11 to both sides. A common mistake is: -4 + 11 = -15 (incorrectly subtracting instead of recognizing this as adding a positive to a negative).

This incorrect simplification may lead them to select Choice A (f(x) = -13x - 11) if they combine errors, or more likely causes them to get confused and abandon their systematic approach.

The Bottom Line:

This problem tests whether students can distinguish between a function and its transformation. The critical moment is recognizing that finding the equation of the shown line is only the FIRST step-you must then work backwards to find f(x) by undoing the "-11" transformation. Students who treat the graph as showing f(x) directly will never arrive at the correct answer.