To graph the function \( y = 4 - \frac{7}{2} x - 1 \), let's first simplify it:
Starting with: \[ y = 4 - \frac{7}{2} x - 1 \] Combine like terms: \[ y = 3 - \frac{7}{2} x \]
Next, we can rewrite it as: \[ y = -\frac{7}{2}x + 3 \] This is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Key Features of the Function:
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Y-Intercept: When \( x = 0 \): \[ y = 3 \] So the y-intercept is \( (0, 3) \).
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Slope: The slope \( m = -\frac{7}{2} \) tells us that for every 2 units we move to the right (positive x-direction), we move 7 units down (negative y-direction).
Finding Another Point:
Let’s find another point by choosing \( x = 2 \): \[ y = -\frac{7}{2}(2) + 3 = -7 + 3 = -4 \] So the point \( (2, -4) \) is also on the line.
Plotting the Points:
Now we can plot the points:
- Y-intercept: \( (0, 3) \)
- Another point: \( (2, -4) \)
Graphing the Line:
- Start at the point \( (0, 3) \) on the y-axis.
- From there, use the slope to find the next point: from \( (0, 3) \) move 2 units to the right to \( x=2 \), and down 7 units to get \( (2, -4) \).
- Draw a straight line through these points extending in both directions.
X-Intercept:
To find the x-intercept where \( y = 0 \): \[ 0 = -\frac{7}{2}x + 3 \] \[ \frac{7}{2}x = 3 \] \[ x = \frac{3 \cdot 2}{7} = \frac{6}{7} \] So the x-intercept is \( \left( \frac{6}{7}, 0 \right) \).
Now your graph should have:
- The y-intercept \( (0, 3) \),
- The x-intercept \( \left( \frac{6}{7}, 0 \right) \),
- Point \( (2, -4) \).
You can draw the graph on a coordinate plane, making sure to label the axes and the intercept points!
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