The graph you provided shows a sinusoidal function, most likely a variation of either a sine or cosine function due to its periodic behavior and shape. Let's analyze it step by step:

Key Features:

1. Amplitude: The graph reaches a maximum value of 5 and a minimum value of 0, so the amplitude (the distance from the middle to the peak) is 2.5. However, since the graph starts at a higher value (not zero), it seems to have a vertical shift.

2. Period: The distance from one peak to the next is $2\pi$, which is typical for sine and cosine functions.

3. Vertical Shift: The graph oscillates between 0 and 5. The middle of this range is $2.5$, suggesting that the function is vertically shifted by 2.5 units upwards.

4. Type of Function: This graph resembles a cosine function, because it reaches a maximum at $x = 0$. A typical cosine function $\cos(x)$ starts at a maximum when $x = 0$.

General Equation:

The form of the cosine function is given by: $y = A \cos(Bx - C) + D$ where:

• $A$ is the amplitude,
• $B$ affects the period ($\text{Period} = \frac{2\pi}{B}$),
• $C$ is the phase shift (horizontal shift),
• $D$ is the vertical shift.

From the graph:

• The amplitude $A = 2.5$,
• The period is $2\pi$, so $B = 1$,
• The vertical shift $D = 2.5$,
• No horizontal shift is observed, so $C = 0$.

Thus, the equation of the graph is: $y = 2.5 \cos(x) + 2.5$

This represents a cosine function with an amplitude of 2.5, a vertical shift of 2.5 units upwards, and a period of $2\pi$.

Would you like more details or have any further questions?

Related Questions:

1. How would the graph look if the amplitude were doubled?
2. What happens to the graph if the period is halved?
3. How does a sine graph compare to this cosine graph?
4. What effect does changing the vertical shift to 0 have on the graph?
5. How would the graph change if there were a phase shift?

Tip:

When analyzing trigonometric graphs, always check for amplitude, period, phase shift, and vertical shift to write the correct equation!