To determine which equation describes the function graphed on the coordinate plane, we follow these steps:

1. Identify the slope:

• The slope of a line is the ratio of the rise (vertical change) to the run (horizontal change) between two points on the line.
• Using the graph, pick two points on the line:
  • Point 1: $(-6, -6)$
  • Point 2: $(0, -2)$
• Compute the slope $m$: $m = \frac{\text{rise}}{\text{run}} = \frac{-2 - (-6)}{0 - (-6)} = \frac{-2 + 6}{6} = \frac{4}{6} = \frac{2}{3}.$

2. Identify the y-intercept:

• The y-intercept is where the line crosses the $y$-axis ($x = 0$).
• From the graph, the line crosses the $y$-axis at $(-2)$. Thus, the y-intercept is $b = -2$.

3. Formulate the equation:

• Using the slope-intercept form of a linear equation: $y = mx + b,$ substitute $m = \frac{2}{3}$ and $b = -2$: $y = \frac{2}{3}x - 2.$

4. Compare with the options:

• The correct equation is $y = \frac{2}{3}x - 2$.

Answer: The correct choice is:

$\boxed{y = \frac{2}{3}x - 2}.$

Would you like further explanation?

5 Relative Questions:

1. How can you find the slope when given a table of points instead of a graph?
2. What is the general form of a linear equation, and how does it relate to the slope-intercept form?
3. How do you convert between slope-intercept form and standard form of a line?
4. Can a line have an undefined slope? If so, what does it look like on a graph?
5. What are the practical uses of linear equations in real-life scenarios?

Tip: To check if an equation matches a graph, substitute a point from the graph into the equation. If it holds true, the equation might describe the graph.