The graph of the linear function \(\mathrm{y=f(x)+19}\) is shown. If c and d are positive constants, which equation could define...

1. TRANSLATE the graph information

Given information:

• The graph shows \(\mathrm{y = f(x) + 19}\) (not \(\mathrm{f(x)}\) directly!)
• The y-intercept appears to be at \(\mathrm{(0, 3)}\)
• The line slopes downward from left to right
• c and d are positive constants

What this tells us:

• We're looking at a transformed version of \(\mathrm{f(x)}\)
• The slope is negative (downward slant)

2. INFER the relationship between the graphs

Key insight: If \(\mathrm{y = f(x) + 19}\) is shown, then:

• Every point on this graph is 19 units HIGHER than the corresponding point on \(\mathrm{y = f(x)}\)
• To find \(\mathrm{f(x)}\), we need to mentally shift this graph DOWN by 19 units
• The slope stays the same (vertical shifts don't affect slope)

3. SIMPLIFY to find the y-intercept of f(x)

The y-intercept of \(\mathrm{y = f(x) + 19}\) is at \(\mathrm{(0, 3)}\), which means:

• When \(\mathrm{x = 0}\): \(\mathrm{f(0) + 19 = 3}\)
• Subtract 19 from both sides: \(\mathrm{f(0) = 3 - 19 = -16}\)

So f(x) has a y-intercept of -16 (negative!)

4. INFER the slope of f(x)

Since vertical translation doesn't change slope:

• \(\mathrm{y = f(x) + 19}\) has negative slope (we can see this)
• Therefore \(\mathrm{f(x)}\) also has negative slope

So f(x) has a negative slope

5. INFER which form matches these requirements

We need an equation where:

• The y-intercept is negative
• The slope is negative
• Using only positive constants c and d

Looking at the structure \(\mathrm{f(x) = (y\text{-}intercept) + (slope)x}\):

• For negative y-intercept from positive d: we need -d
• For negative slope from positive c: we need -cx

Therefore: \(\mathrm{f(x) = -d - cx}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students forget that the graph shows \(\mathrm{y = f(x) + 19}\), not \(\mathrm{f(x)}\) itself. They read the y-intercept as 3 directly from the graph and think \(\mathrm{f(x)}\) must have a positive y-intercept.

Since d is positive, they incorrectly think they need \(\mathrm{+d}\) for the y-intercept. But they correctly see the slope is negative, so they use \(\mathrm{-cx}\).

This may lead them to select Choice B (\(\mathrm{f(x) = d - cx}\)).

Second Most Common Error:

Incomplete INFER reasoning: Students correctly determine that \(\mathrm{f(x)}\) needs a negative y-intercept (\(\mathrm{-d}\)), but they confuse the direction of the slope visually. They might misread the downward slant or think "going down" means the x-coefficient should be positive.

They choose \(\mathrm{-d}\) for the y-intercept but \(\mathrm{+cx}\) for the slope term.

This may lead them to select Choice C (\(\mathrm{f(x) = -d + cx}\)).

The Bottom Line:

This problem tests whether you can track through a transformation (the "+19") and correctly identify BOTH the sign of the y-intercept AND the sign of the slope after accounting for that transformation. The constraint that c and d are positive adds an extra layer - you must use negative signs to create negative values.