The graph of a linear function h is shown in the xy-plane. What is the slope of the line?

1. TRANSLATE the graph information into coordinates

To find the slope, you need two points that the line passes through. Look for places where the line crosses grid intersections clearly.

• First point (y-intercept): The line crosses the y-axis at y = 5, giving us point (0, 5)
• Second point: Following the line, it clearly passes through x = 2, y = 1, giving us point (2, 1)

Key tip: Choose points where the line clearly passes through grid intersections to avoid estimation errors.

2. SIMPLIFY by applying the slope formula

Now use the slope formula with your two points:

\(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)

Substitute \(\mathrm{(x_1, y_1) = (0, 5)}\) and \(\mathrm{(x_2, y_2) = (2, 1)}\):

\(\mathrm{m = \frac{1 - 5}{2 - 0}}\)

\(\mathrm{m = \frac{-4}{2}}\)

\(\mathrm{m = -2}\)

The slope is -2, which makes sense because the line falls steeply from left to right.

Answer: A. -2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misreading coordinates from the graph

Students often misidentify points, especially when:

• They confuse which coordinate is x and which is y
• They miscount grid lines (reading (0, 5) as (0, 4), for example)
• They choose points that don't clearly fall on grid intersections, leading to estimation errors

For example, if a student reads the y-intercept as (0, 4) instead of (0, 5), they would calculate:

\(\mathrm{m = \frac{1 - 4}{2 - 0} = \frac{-3}{2} = -1.5}\)

This doesn't match any answer exactly, leading to confusion and guessing, possibly selecting Choice B (-1/2) as it contains similar values.

Second Most Common Error:

Conceptual confusion about slope formula: Reversing the subtraction order or swapping x and y

Some students might calculate \(\mathrm{\frac{x_2 - x_1}{y_2 - y_1}}\) instead of \(\mathrm{\frac{y_2 - y_1}{x_2 - x_1}}\), giving them:

\(\mathrm{m = \frac{2 - 0}{1 - 5} = \frac{2}{-4} = -\frac{1}{2}}\)

This may lead them to select Choice B (-1/2).

Alternatively, if students swap their points and use \(\mathrm{\frac{2 - 0}{5 - 1} = \frac{2}{4} = \frac{1}{2}}\), they might pick the reciprocal value with wrong sign.

The Bottom Line:

This problem tests whether you can accurately read a graph and correctly apply the slope formula. The key is careful coordinate identification and maintaining the correct order: "rise over run" means (change in y)/(change in x), not the reverse. The negative sign tells you the line is falling, which you can verify by looking at the graph's direction.