To find the function that matches the given graph, we need to analyze its amplitude, period, and vertical shift. Let's go through this step by step:

Step 1: Identify the form of the equation

The form of the function is given as: $y = A \sin(kx) + C \quad \text{or} \quad y = A \cos(kx) + C$ Where:

• $A$ is the amplitude.
• $k$ affects the period (how stretched or compressed the wave is).
• $C$ is the vertical shift.

Step 2: Determine the amplitude $A$

From the graph, we observe that the maximum value is 0 and the minimum value is -4. The amplitude is half the distance between the maximum and minimum values: $A = \frac{0 - (-4)}{2} = 2$ So, $A = 2$.

Step 3: Determine the period and $k$

The period is the length of one complete cycle. From the graph, we see that one cycle occurs between $x = -6$ and $x = 6$, which gives a period of: $\text{Period} = 12$ The period is related to $k$ by the formula: $\text{Period} = \frac{2\pi}{k}$ Thus, we can solve for $k$: $12 = \frac{2\pi}{k} \quad \Rightarrow \quad k = \frac{2\pi}{12} = \frac{\pi}{6}$

Step 4: Determine the vertical shift $C$

The midline of the wave (the horizontal axis around which the sine or cosine oscillates) is at $y = -2$, so $C = -2$.

Step 5: Choose the sine or cosine function

From the graph, it looks like the curve starts at the minimum point when $x = 0$. This behavior corresponds to a cosine function with a phase shift or a sine function that has been reflected. Since the graph looks like a downward-shifted cosine, we use the cosine form.

Final equation

The function that matches the graph is: $y = -2 \cos\left(\frac{\pi}{6}x\right) - 2$

Would you like a deeper explanation or have further questions on this? Here are 5 related questions to explore further:

1. How do we handle phase shifts in sinusoidal functions?
2. Can we also express the same function using a sine form?
3. What happens to the graph if we change the amplitude $A$?
4. How do we calculate the frequency from the period of the wave?
5. What is the significance of the vertical shift $C$ in real-world applications?

Tip: For sinusoidal functions, the period is always given by $\frac{2\pi}{k}$, so larger values of $k$ lead to more compressed waves, while smaller values of $k$ make the wave stretch out.