I am going to describe a graph to you, and I need you to write the equation of the graph to me please: Markings on the x-axis indicate -18°, -9°, 0°, 9°, 18°, 27°, and 36°. There is a dotted line indicating either sine or cosine, with peaks at the -18° and 18° marks and valleys at the 0° and 36° marks. There are zeros at the -9°, 9°, and 27° marks. The zeros are at y=1, the peaks are at y=3, and the valleys are at y=-1. On the actual graph, there are asymptotes at the -9°, 9°, and 27° marks. There are parabolas starting at (-18°, 3) and (18°, 3) going up to infinity and parabolas starting at (0°, -1) and (36°, -1) going down to negative infinity.

Based on your description, the graph you're describing is likely a combination of a sine or cosine function with a tangent function due to the presence of asymptotes and parabolic behavior. Here's the step-by-step interpretation:

### 1. **Identification of Function Type:**
   - **Sine/Cosine Function:** Peaks, valleys, and zeros align with the nature of sine or cosine functions. The peaks at \( -18^\circ \) and \( 18^\circ \) and valleys at \( 0^\circ \) and \( 36^\circ \) suggest a cosine function because cosine typically starts at a maximum.
   - **Tangent Function:** The presence of asymptotes and the parabolic behavior is characteristic of a tangent function.

### 2. **Equation Form:**
   - **Cosine Component:** Considering the zeros and the shape, the cosine function could be modified by an amplitude and phase shift.
   - **Tangent Component:** The parabolas suggest a tangent component, with asymptotes at the locations of zeros of the cosine component.

### 3. **Constructing the Function:**

   Given the periodicity of the cosine function and the points provided:
   - The peaks at \( y = 3 \) and valleys at \( y = -1 \) suggest a shifted and scaled cosine function. We can represent the cosine part as:
     \[
     y = 2\cos\left(\frac{\pi}{18^\circ}x\right) + 1
     \]
   - The tangent function will have asymptotes at the zero points of the cosine function, i.e., at \( -9^\circ \), \( 9^\circ \), and \( 27^\circ \). We can represent this as:
     \[
     y = \tan\left(\frac{\pi}{18^\circ}x\right)
     \]

### 4. **Combined Equation:**
   The combined equation would be:
   \[
   y = \left(2\cos\left(\frac{\pi}{18^\circ}x\right) + 1\right) \cdot \tan\left(\frac{\pi}{18^\circ}x\right)
   \]
   This equation represents the cosine function modified by amplitude and vertical shift, and multiplied by a tangent function to introduce asymptotes and parabolic behavior.

### Final Equation:
\[
y = \left(2\cos\left(\frac{\pi}{18^\circ}x\right) + 1\right) \cdot \tan\left(\frac{\pi}{18^\circ}x\right)
\]

Would you like any further details or explanations? Here are five related questions you might find interesting:

1. How does the period of a trigonometric function change with different coefficients?
2. What are the effects of amplitude and phase shifts on the cosine and sine functions?
3. How do asymptotes relate to the tangent function, and why do they appear?
4. How can we transform a sine function into a cosine function or vice versa?
5. How would the graph change if the function were \( y = \cos(x) \cdot \tan(x) \) without the scaling and shifting?

**Tip:** Remember, when dealing with trigonometric functions, carefully analyze the periodicity, amplitude, and phase shifts to determine the correct form of the function.

Discover how to derive the equation of a graph that combines cosine and tangent functions with peaks, valleys, and asymptotes. This trigonometry problem covers key concepts such as periodic functions, graph analysis, and the behavior of trigonometric functions. Ideal for students in Grades 10-12.

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