To find the y-values for the function \( f(x) = -\cos(x) + 3 \) at the specified \( x \)-values, we will calculate \( f(x) \) for each \( x \):
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For \( x = -\frac{3\pi}{2} \):
\[ f\left(-\frac{3\pi}{2}\right) = -\cos\left(-\frac{3\pi}{2}\right) + 3 \]
Since \( \cos(-\theta) = \cos(\theta) \):
\[ f\left(-\frac{3\pi}{2}\right) = -\cos\left(\frac{3\pi}{2}\right) + 3 = -0 + 3 = 3 \]
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For \( x = -\frac{\pi}{2} \):
\[ f\left(-\frac{\pi}{2}\right) = -\cos\left(-\frac{\pi}{2}\right) + 3 = -\cos\left(\frac{\pi}{2}\right) + 3 = -0 + 3 = 3 \]
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For \( x = \frac{\pi}{2} \):
\[ f\left(\frac{\pi}{2}\right) = -\cos\left(\frac{\pi}{2}\right) + 3 = -0 + 3 = 3 \]
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For \( x = \frac{3\pi}{2} \):
\[ f\left(\frac{3\pi}{2}\right) = -\cos\left(\frac{3\pi}{2}\right) + 3 = -0 + 3 = 3 \]
Now we summarize the calculations:
- \( f\left(-\frac{3\pi}{2}\right) = 3 \)
- \( f\left(-\frac{\pi}{2}\right) = 3 \)
- \( f\left(\frac{\pi}{2}\right) = 3 \)
- \( f\left(\frac{3\pi}{2}\right) = 3 \)
Y-values: All the calculated y-values are 3.
Graph of \( f(x) = -\cos(x) + 3 \)
The function \( -\cos(x) + 3 \) takes the basic cosine wave, inverts it (flips it upside down) and then shifts it upwards by 3 units.
- The cosine function oscillates between -1 and 1. Thus, \( -\cos(x) \) oscillates between -1 and 1 as well.
- Adding 3 shifts this range to oscillate between 2 and 4.
So the graph will:
- Have a midline at \( y = 3 \)
- Reach a maximum value of \( 4 \) (when \( \cos(x) = -1 \))
- Reach a minimum value of \( 2 \) (when \( \cos(x) = 1 \))
Characteristics:
- The peaks of the graph occur when \( x \) is at \( 2n\pi \) (where \( n \) is an integer).
- The troughs occur at \( (2n+1)\pi \) (where \( n \) is an integer).
Therefore, the graph is a cosine wave that oscillates horizontally and has a vertical translation that centers it around y = 3.
If you would like to see a specific graphical representation or need more details about the graph, please let me know!