Analyzing the graph:

The provided graph represents a cosine function. To write the equation in the form $f(x) = A \cos(Bx + C) + D$, we analyze the graph based on its amplitude, period, phase shift, and vertical shift.

Step 1: Amplitude ($A$)

The amplitude is the distance from the midline to the maximum (or minimum). From the graph:

• Maximum value = $2$
• Minimum value = $-2$
• Midline (average of max and min) = $0$

Thus, the amplitude is: $A = 2 - 0 = 2$

Step 2: Period ($B$)

The period is the length of one complete cycle. From the graph, the cosine repeats every $2\pi$. The formula for the period is: $\text{Period} = \frac{2\pi}{B}$ Setting the period equal to $2\pi$: $\frac{2\pi}{B} = 2\pi \implies B = 1$

Step 3: Phase Shift ($C$)

The phase shift is determined by where the graph starts relative to the standard cosine function $\cos(x)$.

In the graph, the cosine starts at $(x, f(x)) = (0, 2)$, which matches the standard cosine function. Hence: $C = 0$

Step 4: Vertical Shift ($D$)

The vertical shift moves the graph up or down. From the graph, the midline is at $y = 0$, so there is no vertical shift: $D = 0$

Final Equation:

Substituting $A = 2$, $B = 1$, $C = 0$, and $D = 0$ into the general form $f(x) = A \cos(Bx + C) + D$, the equation becomes: $f(x) = 2 \cos(x)$

Would you like to see further steps, confirm details, or analyze a related graph? Let me know!

Related Questions:

1. What happens to the graph if the amplitude is doubled?
2. How would a phase shift of $\frac{\pi}{2}$ affect the equation?
3. What is the impact of adding a vertical shift $D \neq 0$?
4. How can the period be modified by changing $B$?
5. What would the equation look like if the graph were a sine function instead?

Tip:

To quickly find the period of a trigonometric graph, measure the distance between two consecutive peaks (or troughs) on the x-axis.