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

### Gradient-Intercept Method for \( y = \frac{2}{3}x - 2 \)

To sketch the graph and make a table for the line \( y = \frac{2}{3}x - 2 \), let's go step by step.

#### Step 1: Identify the slope (gradient) and the y-intercept.
- **Slope (Gradient):** \( m = \frac{2}{3} \)
- **Y-Intercept:** The line crosses the y-axis at \( y = -2 \), so the intercept is \( (0, -2) \).

#### Step 2: Plot the y-intercept.
- Start by plotting the point \( (0, -2) \) on the graph. This is where the line crosses the y-axis.

#### Step 3: Use the slope to find additional points.
The slope \( m = \frac{2}{3} \) tells us that for every 3 units you move to the right (increase in \( x \)), the \( y \)-value increases by 2 units. You can also go in the reverse direction (left by 3, down by 2).

- From \( (0, -2) \), move 3 units to the right and 2 units up to get the point \( (3, 0) \).
- You can also go in the reverse direction: from \( (0, -2) \), move 3 units to the left and 2 units down to get the point \( (-3, -4) \).

#### Step 4: Draw the line.
Connect the points \( (0, -2) \), \( (3, 0) \), and \( (-3, -4) \) with a straight line. This is the graph of the equation \( y = \frac{2}{3}x - 2 \).

#### Step 5: Make a table of values.
Here’s a table of values to correspond with the line:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-6 & -6 \\
-3 & -4 \\
0 & -2 \\
3 & 0 \\
6 & 2 \\
\hline
\end{array}
\]

#### Visualization:
You can sketch the graph based on the points in the table. It will be a straight line, sloping upwards with a gentle rise (since the slope is less than 1).

Would you like any further clarification on how this method works, or do you have any questions?

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Here are 5 follow-up questions related to this topic:
1. How do you find the slope if the equation is not in slope-intercept form?
2. What happens to the graph when the slope is negative?
3. Can you determine the x-intercept algebraically from the equation \( y = \frac{2}{3}x - 2 \)?
4. How do you interpret the slope in real-world problems?
5. What changes would occur if the y-intercept were positive instead?

**Tip:** Always check your table values with the original equation to ensure accuracy when sketching graphs!

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. This problem covers linear equations and graphing techniques. Suitable for students in Grades 7-9.

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