https://file.aiquickdraw.com/mathbot/file/1728705470034erl27hko.jpg

The graph in the image is of a cosine function that is vertically flipped and shifted. Let's break down the information provided by the graph and construct the equation in the form \( y = A \cos(\omega x) \).

1. **Amplitude (A):** The graph reaches a maximum of 10 and a minimum of -10, which means the amplitude is 10. The negative sign indicates that the graph is reflected vertically, so \( A = -10 \).

2. **Period (T):** The period is the horizontal distance required for the graph to complete one full cycle. From the graph, it completes one full cycle from \( x = -2\pi \) to \( x = 6\pi \). Thus, the period is \( 6\pi - (-2\pi) = 8\pi \).

   The period \( T \) of a cosine function is related to \( \omega \) by the formula:
   \[
   T = \frac{2\pi}{\omega}
   \]
   Setting \( T = 8\pi \), we get:
   \[
   8\pi = \frac{2\pi}{\omega}
   \]
   Solving for \( \omega \):
   \[
   \omega = \frac{2\pi}{8\pi} = \frac{1}{4}
   \]

3. **Equation:** Now, using these values, the equation of the cosine function is:
   \[
   y = -10 \cos\left(\frac{1}{4}x\right)
   \]

Thus, the equation for the graph is:
\[
y = -10 \cos\left(\frac{1}{4}x\right)
\]

Would you like more details or have any questions about this? Here are 5 related questions to expand your understanding:

1. How does the amplitude affect the graph of a trigonometric function?
2. What does it mean when a cosine function is reflected vertically?
3. How can you determine the phase shift of a cosine or sine function?
4. What are the differences between sine and cosine functions in terms of their graphs?
5. How do changes in \( \omega \) affect the frequency of a cosine or sine function?

**Tip:** The phase shift of a trigonometric function occurs when the graph is shifted horizontally. Keep an eye on this when analyzing equations like \( A \sin(\omega x + \phi) \).

Find an equation for the graph shown, in the form y = A sin(ωx) or y = A cos(ωx).

Learn how to determine the equation of a cosine function from a graph with a specific amplitude and period. In this case, the function has an amplitude of 10, a period of 8π, and is vertically flipped. This is suitable for students in Grades 10-12 studying trigonometric functions.

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