The scatterplot shows the relationship between two variables, x and y. A line of best fit is also shown. Which...

1. VISUALIZE the line and identify two clear points

When finding the slope of a line from a graph, you need two points that lie on the line (not the scattered data points).

• Look for where the line of best fit crosses the axes - these give the cleanest coordinates:
  • y-intercept: The line crosses the y-axis at (0, 14)
  • x-intercept: The line crosses the x-axis at (13, 0)

Key insight: Use the line itself, not the individual data points scattered around it.

2. TRANSLATE the visual coordinates into mathematical form

Now that we've identified our two points from the graph:

• Point 1: \(\mathrm{(x_1, y_1) = (0, 14)}\)
• Point 2: \(\mathrm{(x_2, y_2) = (13, 0)}\)

3. INFER which formula to apply

Since we need the slope and have two points, we'll use the slope formula:

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

4. SIMPLIFY by substituting and calculating

Substitute the coordinates:

\(\mathrm{m = \frac{0 - 14}{13 - 0}}\)

\(\mathrm{m = \frac{-14}{13}}\)

To compare with the answer choices, convert to decimal (use calculator):

\(\mathrm{m \approx -1.077}\)

5. APPLY CONSTRAINTS to select the closest answer

Looking at the choices:

• A. -3.3 (much steeper than our calculated value)
• B. -1.1 (very close to -1.077)
• C. 1.1 (positive, but our slope is negative)
• D. 3.3 (wrong sign and wrong magnitude)

Answer: B (-1.1)

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Why Students Usually Falter on This Problem

Most Common Error Path #1:

VISUALIZE error - Misreading coordinates: Students may miscount grid squares or misidentify where the line crosses the axes. For example:

• Reading the y-intercept as (0, 13) instead of (0, 14)
• Reading the x-intercept as (12, 0) or (14, 0) instead of (13, 0)

If a student reads the points as (0, 13) and (12, 0), they would calculate:

\(\mathrm{m = \frac{0 - 13}{12 - 0} = \frac{-13}{12} \approx -1.08}\)

This still leads to Choice B (-1.1) by luck, but if they read (0, 14) and (4, 0):

\(\mathrm{m = \frac{0 - 14}{4 - 0} = \frac{-14}{4} = -3.5}\)

This could lead them to select Choice A (-3.3).

Most Common Error Path #2:

TRANSLATE error - Reversing the slope formula: Students may confuse rise over run and calculate \(\mathrm{\frac{x_2 - x_1}{y_2 - y_1}}\) instead:

\(\mathrm{m = \frac{13 - 0}{0 - 14} = \frac{13}{-14} = \frac{-13}{14} \approx -0.93}\)

This value doesn't match any answer choice closely, causing confusion and potentially leading to guessing or selecting Choice B (-1.1) as the "closest negative value near -1."

Most Common Error Path #3:

SIMPLIFY error - Dropping the negative sign: Students may correctly calculate \(\mathrm{|\frac{-14}{13}| = \frac{14}{13} \approx 1.1}\) but forget that the slope is negative because the line is decreasing.

This leads them to select Choice C (1.1) instead of the correct Choice B (-1.1).

The Bottom Line:

This problem tests whether students can accurately extract coordinates from a visual representation and correctly apply the slope formula. The most critical skills are careful graph reading and maintaining proper sign conventions throughout the calculation.