What is an equation of the graph shown?

1. TRANSLATE the graph information

Looking at the graph carefully, I need to identify points where the line passes through grid intersections:

• Given: A line graphed on a coordinate plane
• The line crosses the y-axis at (0, -8) - this is the y-intercept
• The line crosses the x-axis at (-8, 0) - this is the x-intercept
• Both points are clearly on grid intersections, making them reliable

2. INFER the solution strategy

To find the equation, I'll use slope-intercept form \(\mathrm{y = mx + b}\) because:

• I can directly read the y-intercept from where the line crosses the y-axis: \(\mathrm{b = -8}\)
• I can calculate the slope m using the two points I identified
• This is the most efficient approach since b is immediately visible

3. SIMPLIFY to calculate the slope

Using the slope formula with points (0, -8) and (-8, 0):

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

\(\mathrm{m = \frac{0 - (-8)}{-8 - 0}}\)

\(\mathrm{m = \frac{0 + 8}{-8}}\)

\(\mathrm{m = \frac{8}{-8}}\)

\(\mathrm{m = -1}\)

⚠️ Be careful here! Subtracting a negative means adding: \(\mathrm{0 - (-8) = 0 + 8 = 8}\)

4. SIMPLIFY to write the final equation

Now substitute \(\mathrm{m = -1}\) and \(\mathrm{b = -8}\) into \(\mathrm{y = mx + b}\):

• \(\mathrm{y = (-1)x + (-8)}\)
• \(\mathrm{y = -x - 8}\)

This matches Choice C.

Answer: C. y = -x - 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution - Sign error in slope calculation: Students make mistakes when handling the subtraction of negative numbers in the slope formula.

When calculating \(\mathrm{m = \frac{0 - (-8)}{-8 - 0}}\), students might:

• Incorrectly simplify \(\mathrm{0 - (-8)}\) as \(\mathrm{0 - 8 = -8}\) instead of \(\mathrm{0 + 8 = 8}\)
• This gives \(\mathrm{m = \frac{-8}{-8} = 1}\) instead of \(\mathrm{m = \frac{8}{-8} = -1}\)
• With the correct \(\mathrm{b = -8}\), they write \(\mathrm{y = x - 8}\)

This may lead them to select Choice B (y = x - 8)

Second Most Common Error:

Weak SIMPLIFY execution - Arithmetic/calculation error with slope: Students confuse the rise and run or make other calculation errors when finding the slope.

For example, they might:

• Count incorrectly on the graph (counting "run" as 4 instead of 8)
• Mix up which direction is positive/negative
• Calculate \(\mathrm{\frac{8}{-4} = -2}\) instead of \(\mathrm{\frac{8}{-8} = -1}\)
• With \(\mathrm{b = -8}\), this gives \(\mathrm{y = -2x - 8}\)

This may lead them to select Choice A (y = -2x - 8)

Third Most Common Error:

Poor TRANSLATE skill: Students misread one or both coordinates from the graph, especially with negative values on both axes.

If students misidentify the points but still apply the correct process, they'll arrive at an incorrect equation. However, since all answer choices have -8 as the y-intercept, a misread y-intercept would cause confusion rather than leading to a specific wrong answer. This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can accurately extract information from a visual representation and carefully handle negative numbers throughout their calculations. The key challenge is maintaining precision with signs—both when reading the graph and when performing the slope calculation. One small sign error propagates through to an incorrect final answer.