Math Problem Statement

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

Solution

Gradient-Intercept Method for y=23x2y = \frac{2}{3}x - 2

To sketch the graph and make a table for the line y=23x2y = \frac{2}{3}x - 2, let's go step by step.

Step 1: Identify the slope (gradient) and the y-intercept.

  • Slope (Gradient): m=23m = \frac{2}{3}
  • Y-Intercept: The line crosses the y-axis at y=2y = -2, so the intercept is (0,2)(0, -2).

Step 2: Plot the y-intercept.

  • Start by plotting the point (0,2)(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=23m = \frac{2}{3} tells us that for every 3 units you move to the right (increase in xx), the yy-value increases by 2 units. You can also go in the reverse direction (left by 3, down by 2).

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

Step 4: Draw the line.

Connect the points (0,2)(0, -2), (3,0)(3, 0), and (3,4)(-3, -4) with a straight line. This is the graph of the equation y=23x2y = \frac{2}{3}x - 2.

Step 5: Make a table of values.

Here’s a table of values to correspond with the line:

\hline x & y \\ \hline -6 & -6 \\ -3 & -4 \\ 0 & -2 \\ 3 & 0 \\ 6 & 2 \\ \hline \end{array}$$ #### Visualization: You can sketch the graph based on the points in the table. It will be a straight line, sloping upwards with a gentle rise (since the slope is less than 1). Would you like any further clarification on how this method works, or do you have any questions? --- Here are 5 follow-up questions related to this topic: 1. How do you find the slope if the equation is not in slope-intercept form? 2. What happens to the graph when the slope is negative? 3. Can you determine the x-intercept algebraically from the equation $$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!

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Graphing

Formulas

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Theorems

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Suitable Grade Level

Grades 7-9