Which of the following is an equation of the graph shown in the xy-plane above?

1. TRANSLATE the graph information

• TRANSLATE the y-intercept from the graph:
  • Look where the line crosses the y-axis (the vertical axis)
  • This occurs at the point \(\mathrm{(0, -1)}\)
  • The y-intercept \(\mathrm{b = -1}\)

2. TRANSLATE a second point from the graph

• Choose another clear point where the line passes through a grid intersection:
  • The line clearly passes through \(\mathrm{(4, -2)}\)
  • We now have two points: \(\mathrm{(0, -1)}\) and \(\mathrm{(4, -2)}\)

3. INFER that we need to calculate the slope

• To write the equation \(\mathrm{y = mx + b}\), we need both \(\mathrm{m}\) (slope) and \(\mathrm{b}\) (y-intercept)
• We already have \(\mathrm{b = -1}\)
• We need to find \(\mathrm{m}\) using the two points we identified

4. SIMPLIFY to calculate the slope

• Apply the slope formula:
  • \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
  • \(\mathrm{m = \frac{-2 - (-1)}{4 - 0}}\)
  • \(\mathrm{m = \frac{-2 + 1}{4}}\)
  • \(\mathrm{m = -\frac{1}{4}}\)

5. INFER how to construct the complete equation

• We have:
  • slope \(\mathrm{m = -\frac{1}{4}}\)
  • y-intercept \(\mathrm{b = -1}\)
• In slope-intercept form: \(\mathrm{y = mx + b}\)
• Substitute our values: \(\mathrm{y = -\frac{1}{4}x + (-1)}\)
• Simplified: \(\mathrm{y = -\frac{1}{4}x - 1}\)

6. APPLY CONSTRAINTS to match with answer choices

• Compare our equation \(\mathrm{y = -\frac{1}{4}x - 1}\) with the choices
• This exactly matches Choice A

Answer: A. y = -1/4 x - 1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Incorrectly calculating the slope, particularly with the negative numbers

Students may calculate:

• \(\mathrm{m = \frac{-1 - (-2)}{0 - 4} = \frac{1}{-4} = -\frac{1}{4}}\) (correct by reversing the order)
• OR \(\mathrm{m = \frac{-2 - (-1)}{0 - 4} = \frac{-1}{-4} = \frac{1}{4}}\) (wrong - forgot one negative)

If they get \(\mathrm{m = -4}\) by computing \(\mathrm{\frac{4}{-1}}\) or making other calculation errors, they might select Choice D (y = -4x - 1).

Second Most Common Error:

Poor INFER reasoning: Confusing which calculated value represents the slope versus the y-intercept

After correctly calculating \(\mathrm{slope = -\frac{1}{4}}\), students might incorrectly think:

• "The slope is -1 and the y-intercept is -1/4"
• This backwards thinking leads them to select Choice C (y = -x - 1/4)

Third Most Common Error:

Inadequate TRANSLATE execution: Misreading coordinates from the graph

Students might:

• Misidentify the y-intercept as \(\mathrm{(0, -4)}\) instead of \(\mathrm{(0, -1)}\)
• Or choose poor points that don't lie exactly on grid intersections
• This can lead to \(\mathrm{slope = -1}\) with y-intercept = -4, selecting Choice B (y = -x - 4)

The Bottom Line:

This problem tests your ability to accurately extract visual information (coordinates) from a graph and then systematically apply the slope formula. The key is being methodical: first identify the y-intercept clearly, then choose a second clear point, calculate slope carefully with attention to negative signs, and finally match your values correctly to the slope-intercept form.