To determine which equations best represent the graph, we need to analyze the line on the coordinate plane. Let's go through the key points to understand the slope and y-intercept of this line.

Step 1: Identifying the Slope

From the graph:

• The line passes through points $(0, -3)$ and $(2, 3)$.
• Using these points, we can calculate the slope ($m$) as follows:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - (-3)}{2 - 0} = \frac{6}{2} = 3$

Step 2: Identifying the Y-Intercept

The y-intercept occurs when $x = 0$. Looking at the graph, we see that the line crosses the y-axis at $-3$. Thus, the y-intercept $b$ is $-3$.

Step 3: Writing the Equation

With a slope of $3$ and a y-intercept of $-3$, the equation of the line in slope-intercept form ($y = mx + b$) is:

$y = 3x - 3$

Step 4: Selecting the Correct Equations

We now look for equations equivalent to $y = 3x - 3$ among the options:

• Option A: $y - 3 = 3(x - 2)$

  • Expanding: $y - 3 = 3x - 6$
  • $y = 3x - 3$ (This matches our equation.)

• Option D: $y = 3x - 3$

  • This is exactly the equation we derived.

Thus, the correct answers are:

A and D

Would you like a detailed explanation of any step, or do you have further questions?

Here are five related questions for further exploration:

1. How can we find the equation of a line given two points?
2. What is the significance of slope in linear equations?
3. How do transformations affect the equation of a line?
4. How can we convert between different forms of a line equation?
5. What are the characteristics of parallel lines on a graph?

Tip: When given a graph, always check the slope and y-intercept to find the line’s equation.