The graph of the linear function \(\mathrm{y = -2f(x)}\) is shown. If a and b are positive constants, which equation...

1. TRANSLATE the graph into an equation

The first step is reading the graph carefully. Notice two things:

• The graph shows a straight line
• The title says the graph represents \(\mathrm{y = -2f(x)}\) (not \(\mathrm{f(x)}\) itself!)

Identify clear points on the line:

• At \(\mathrm{x = 0}\), the line crosses at \(\mathrm{y = -2}\) → Point \(\mathrm{(0, -2)}\)
• At \(\mathrm{x = 1}\), the line crosses at \(\mathrm{y = 0}\) → Point \(\mathrm{(1, 0)}\)
• At \(\mathrm{x = 3}\), the line is at \(\mathrm{y = 4}\) → Point \(\mathrm{(3, 4)}\)

2. SIMPLIFY to find the equation of the graphed line

Calculate the slope using any two points:

• Slope = \(\frac{0 - (-2)}{1 - 0} = \frac{2}{1} = 2\)

Identify the y-intercept from the graph:

• The line crosses the y-axis at -2

Write the equation in slope-intercept form:

• \(\mathrm{y = 2x - 2}\)

This is the equation of the line shown in the graph.

3. INFER the relationship between the graph and f(x)

Here's the crucial insight: The problem tells us the graph shows \(\mathrm{y = -2f(x)}\), not \(\mathrm{f(x)}\).

So we have:

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

We need to find what \(\mathrm{f(x)}\) equals.

4. SIMPLIFY to solve for f(x)

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

Divide both sides by -2:

\(\mathrm{f(x) = \frac{2x - 2}{-2}}\)

Simplify the right side:

\(\mathrm{f(x) = \frac{2x}{-2} - \frac{2}{-2}}\)

\(\mathrm{f(x) = -x + 1}\)

5. INFER which answer choice matches this form

Now compare \(\mathrm{f(x) = -x + 1}\) with the answer choices, remembering that a and b are positive constants:

• (A) \(\mathrm{f(x) = -ax - b}\): If a and b are positive, this gives \(\mathrm{-x - (positive\ number) = -x - b}\). Our function has +1, not -1. ✗
• (B) \(\mathrm{f(x) = ax - b}\): If a and b are positive, this gives \(\mathrm{+x - (positive\ number)}\). Our function starts with -x, not +x. ✗
• (C) \(\mathrm{f(x) = -ax + b}\): If a and b are positive, this gives \(\mathrm{-x + (positive\ number)}\). With \(\mathrm{a = 1}\) and \(\mathrm{b = 1}\): \(\mathrm{f(x) = -1 \cdot x + 1 = -x + 1}\) ✓
• (D) \(\mathrm{f(x) = ax + b}\): If a and b are positive, this gives \(\mathrm{+x + (positive\ number)}\). Our function has -x and this has +x. ✗

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misread what the graph represents and think it shows \(\mathrm{f(x)}\) directly, rather than \(\mathrm{y = -2f(x)}\).

If they believe the graph shows \(\mathrm{f(x)}\), they would conclude:

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

Then looking at the answer choices with positive a and b:

• This would match \(\mathrm{f(x) = ax - b}\) with \(\mathrm{a = 2, b = 2}\)

This leads them to select Choice B (\(\mathrm{f(x) = ax - b}\)).

Second Most Common Error:

Poor INFER reasoning about signs: Students correctly find \(\mathrm{f(x) = -x + 1}\) but then get confused about which answer form this matches.

The confusion often centers on: "Does \(\mathrm{-x + 1}\) match \(\mathrm{-ax + b}\) or \(\mathrm{-ax - b}\)?"

They might think:

• "Well, \(\mathrm{-x + 1}\) could be written as \(\mathrm{-x - (-1)}\), so maybe it's \(\mathrm{-ax - b}\) with \(\mathrm{b = -1}\)"
• But the problem states b is a POSITIVE constant, so b cannot be -1

This confusion might cause them to guess between choices (A) and (C), potentially selecting Choice A (\(\mathrm{-ax - b}\)).

The Bottom Line:

This problem requires careful attention to what the graph actually represents (a transformation of f, not f itself) and precise algebraic manipulation to work backward from \(\mathrm{-2f(x)}\) to \(\mathrm{f(x)}\). The answer choice matching requires understanding how signs work with the constraint that a and b are positive.