Question

Using f(x)=−cosx+3 , find the y-values in the second row by substituting each x-value into the function. Which of the following is the graph of f(x) ? x −3π/2 −π/2 π/2 3π/2 f(x)
what does teh graph look like?

Answer 1

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 \( x \):

1. For \( x = -\frac{3\pi}{2} \):

\[
f\left(-\frac{3\pi}{2}\right) = -\cos\left(-\frac{3\pi}{2}\right) + 3
\]

Since \( \cos(-\theta) = \cos(\theta) \):

\[
f\left(-\frac{3\pi}{2}\right) = -\cos\left(\frac{3\pi}{2}\right) + 3 = -0 + 3 = 3
\]

2. For \( x = -\frac{\pi}{2} \):

\[
f\left(-\frac{\pi}{2}\right) = -\cos\left(-\frac{\pi}{2}\right) + 3 = -\cos\left(\frac{\pi}{2}\right) + 3 = -0 + 3 = 3
\]

3. For \( x = \frac{\pi}{2} \):

\[
f\left(\frac{\pi}{2}\right) = -\cos\left(\frac{\pi}{2}\right) + 3 = -0 + 3 = 3
\]

4. 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!