To fill in the table for the linear equation \( y = \frac{1}{4}x - 3 \), we will calculate the value of \( y \) for each given \( x \).
Calculations
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For \( x = -4 \): \[ y = \frac{1}{4}(-4) - 3 = -1 - 3 = -4 \]
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For \( x = -2 \): \[ y = \frac{1}{4}(-2) - 3 = -\frac{1}{2} - 3 = -\frac{1}{2} - \frac{6}{2} = -\frac{7}{2} = -3.5 \]
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For \( x = 0 \): \[ y = \frac{1}{4}(0) - 3 = 0 - 3 = -3 \]
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For \( x = 4 \): \[ y = \frac{1}{4}(4) - 3 = 1 - 3 = -2 \]
Completed Table
Now, let's summarize the values in the table:
| X | Y | |-----|-------| | -4 | -4 | | -2 | -3.5 | | 0 | -3 | | 4 | -2 |
Graphing the Linear Equation
To graph the line \( y = \frac{1}{4}x - 3 \):
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Plot the points from the table:
- Point (-4, -4)
- Point (-2, -3.5)
- Point (0, -3)
- Point (4, -2)
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Draw the line through the points.
Graphing Instructions
Given that your graph has a horizontal scale of 5 units and a vertical scale of 5 units, you can add appropriate labels along the axes to accommodate the points you calculated. Make sure the four quadrants are represented, with the considering the negative values for \( y \), especially for the plotted points.
The slope of \( \frac{1}{4} \) indicates that for every 4 units you move horizontally to the right (increasing \( x \)), the \( y \) value increases by 1 unit, hence depicting a gentle upward slope.
If you are drawing on graph paper or using graphing software, ensure that the axes extend adequately to display your specified points from the table.
Final Thoughts
You can now create a clear graph representing the linear equation with the points plotted, resulting in a straight line as indicated by the equation.