To plot the function \( y = -\frac{1}{2}x^3 \), we need to calculate the corresponding \( y \) values for five different \( x \) values — one at \( x = 0 \), two negative, and two positive.
Let's calculate the points:
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For \( x = 0 \): \[ y = -\frac{1}{2}(0)^3 = 0 \quad \Rightarrow \quad (0, 0) \]
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For \( x = -1 \): \[ y = -\frac{1}{2}(-1)^3 = -\frac{1}{2}(-1) = \frac{1}{2} \quad \Rightarrow \quad (-1, \frac{1}{2}) \]
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For \( x = -2 \): \[ y = -\frac{1}{2}(-2)^3 = -\frac{1}{2}(-8) = 4 \quad \Rightarrow \quad (-2, 4) \]
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For \( x = 1 \): \[ y = -\frac{1}{2}(1)^3 = -\frac{1}{2}(1) = -\frac{1}{2} \quad \Rightarrow \quad (1, -\frac{1}{2}) \]
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For \( x = 2 \): \[ y = -\frac{1}{2}(2)^3 = -\frac{1}{2}(8) = -4 \quad \Rightarrow \quad (2, -4) \]
Now, we have the following five points:
- \( (0, 0) \)
- \( (-1, \frac{1}{2}) \)
- \( (-2, 4) \)
- \( (1, -\frac{1}{2}) \)
- \( (2, -4) \)
To visualize the graph, you would plot these points on the coordinate system, and you'll see the nature of the cubic function \( y = -\frac{1}{2}x^3 \), which is decreasing for positive \( x \) and increasing for negative \( x \).
If you're using specific graphing software or an online graphing tool, you would input this function to see the overall graph and then add the calculated points. If you have a button for graphing the function, be sure to highlight or click it after inputting the function.