To determine the correct equation for the line shown in the graph, we need to analyze two main properties of the line:

1. Slope (m) - The steepness of the line.
2. Y-intercept (b) - The point where the line crosses the y-axis.

Step 1: Identifying the slope

Looking at the graph, the line slopes downward from left to right, meaning it has a negative slope.

We can calculate the slope by using two points on the line. The line crosses at (-2, 6) and (2, -6). The formula for slope is: $m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$ Substitute the points (-2, 6) and (2, -6): $m = \frac{{-6 - 6}}{{2 - (-2)}} = \frac{{-12}}{{4}} = -3$ So, the slope is -3.

Step 2: Identifying the y-intercept

The line crosses the y-axis at $y = 0$. Therefore, the y-intercept (b) is 0.

Step 3: Equation of the line

The slope-intercept form of a line is given by: $y = mx + b$ Using the values $m = -3$ and $b = 2$, the equation of the line is: $y = -3x + 2$

Therefore, the correct answer is A: $y = -3x + 2$.

Would you like further clarification, or do you have any questions?

Related Questions:

1. How do you determine the slope of a line from a graph?
2. What is the slope-intercept form of a line?
3. How do you find the y-intercept from a graph?
4. Can the slope of a line be positive and negative? How does it affect the graph?
5. How can we use two points on a line to find the equation?

Tip:

When graphing a line, always check the slope direction (upward or downward) to determine if it's positive or negative!