Question

which of the following is an equation for the graph?

a. (2 sin(2x + \frac{pi}{2}) + 2)
b. (sin(2x + pi) + 2)
c. (2 sin(2x - pi) - 2)
d. (2 sin(2x - \frac{pi}{2}) + 2)

Explanation:

⚡ Using what you learned: Graphing Trigonometric Functions

Step 1: Identify Midline and Amplitude

The maximum value of the graph is \(3\) and the minimum value is \(1\).

$$\text{Midline } (d) = \frac{\text{Maximum} + \text{Minimum}}{2} = \frac{3 + 1}{2} = 2$$

$$\text{Amplitude } (a) = \frac{\text{Maximum} - \text{Minimum}}{2} = \frac{3 - 1}{2} = 1$$

Since the amplitude is \(1\) and the midline is \(y = 2\), the equation must have the form:

$$y = \sin(2x - c) + 2$$

This eliminates options A, C, and D, which all have an amplitude of \(2\).

Step 2: Verify with a Point

Let's test a point from the graph on the remaining option, B: \(y = \sin(2x + \pi) + 2\).

At \(x = 0\), the graph passes through \(y = 2\).

Substitute \(x = 0\) into option B:

$$y = \sin(2(0) + \pi) + 2 = \sin(\pi) + 2 = 0 + 2 = 2$$

This matches the graph perfectly.

Answer:

B. \(\sin(2x + \pi) + 2\)