Using the gradient intercept method, sketch and make a table of y= 2/3 x -2 pls make the table from -3 to 3 

Sure! Let's complete the task of creating the table of values for the equation \( y = \frac{2}{3}x - 2 \) with \( x \) values ranging from -3 to 3. We will also sketch the graph using the gradient-intercept method.

### Step 1: Identify the slope and y-intercept.
- **Slope (Gradient):** \( m = \frac{2}{3} \)
- **Y-Intercept:** \( (0, -2) \)

### Step 2: Calculate the table of values.

We will substitute values of \( x \) from -3 to 3 into the equation \( y = \frac{2}{3}x - 2 \) and compute the corresponding \( y \)-values.

\[
y = \frac{2}{3}x - 2
\]

| \( x \) | \( y = \frac{2}{3}x - 2 \) |
|--------|----------------------------|
| -3     | \( y = \frac{2}{3}(-3) - 2 = -2 - 2 = -4 \) |
| -2     | \( y = \frac{2}{3}(-2) - 2 = -\frac{4}{3} - 2 = -\frac{10}{3} \approx -3.33 \) |
| -1     | \( y = \frac{2}{3}(-1) - 2 = -\frac{2}{3} - 2 = -\frac{8}{3} \approx -2.67 \) |
| 0      | \( y = \frac{2}{3}(0) - 2 = -2 \) |
| 1      | \( y = \frac{2}{3}(1) - 2 = \frac{2}{3} - 2 = -\frac{4}{3} \approx -1.33 \) |
| 2      | \( y = \frac{2}{3}(2) - 2 = \frac{4}{3} - 2 = -\frac{2}{3} \approx -0.67 \) |
| 3      | \( y = \frac{2}{3}(3) - 2 = 2 - 2 = 0 \) |

### Step 3: Create the table of values.
Here’s the table of values for \( x \) ranging from -3 to 3:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3 & -4 \\
-2 & -3.33 \\
-1 & -2.67 \\
0  & -2 \\
1  & -1.33 \\
2  & -0.67 \\
3  & 0 \\
\hline
\end{array}
\]

### Step 4: Sketch the graph.

- Plot the points from the table: \( (-3, -4) \), \( (-2, -3.33) \), \( (-1, -2.67) \), \( (0, -2) \), \( (1, -1.33) \), \( (2, -0.67) \), and \( (3, 0) \).
- Draw a straight line through these points, since the equation represents a straight line with a slope of \( \frac{2}{3} \).

The line will slope upwards, crossing the y-axis at \( (0, -2) \), with a consistent rise of 2 units for every 3 units moved to the right along the x-axis.

Would you like me to provide a visual sketch of the graph?

---

Here are 5 follow-up questions related to this problem:
1. How can you verify the accuracy of the points you plotted on the graph?
2. How does the slope affect the steepness of the line?
3. Can you predict what would happen if the slope were \( \frac{3}{2} \) instead of \( \frac{2}{3} \)?
4. What happens to the graph if the y-intercept changes to a positive value?
5. How do you find the x-intercept from the equation \( y = \frac{2}{3}x - 2 \)?

**Tip:** When plotting points from a table, always label your axes and points to avoid confusion during the graphing process.

Learn how to sketch the graph and create a table of values for the linear equation y = (2/3)x - 2 using the gradient-intercept method. Includes step-by-step calculation of values from -3 to 3 and graphical representation. Suitable for students in Grades 7-9.

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