the graph of a trigonometric function is shown.\nwhat is the amplitude of this function?

Explanation:

Step1: Recall amplitude formula

Amplitude of a trigonometric function is the distance from the midline to the maximum (or minimum) value. The formula is ( \text{Amplitude} = \frac{\text{Max Value} - \text{Min Value}}{2} ).

Step2: Identify max and min values

From the graph, the maximum ( y )-value (peak) is ( 3 ), and the minimum ( y )-value (trough) is ( -1 ).

Step3: Calculate amplitude

Substitute into the formula: ( \frac{3 - (-1)}{2} = \frac{4}{2} = 2 ). Wait, wait, no—wait, maybe midline first. Wait, midline is the average of max and min. Midline ( = \frac{3 + (-1)}{2} = \frac{2}{2} = 1 ). Then amplitude is distance from midline to max: ( 3 - 1 = 2 )? Wait, no, wait the graph: looking at the y-axis, the midline—wait, the graph crosses the y-axis at (0,1), and the peaks are at y=3, troughs at y=-1. So midline is y=1. Then amplitude is |3 - 1| = 2? Wait, no, wait maybe I made a mistake. Wait, let's check again. Wait, the standard amplitude is (max - min)/2. Max is 3, min is -1. So (3 - (-1))/2 = 4/2 = 2. Wait, but wait the graph: when x=0, y=1, which is the midline? Wait, no, maybe the function is a sine or cosine function shifted? Wait, no, the amplitude is the maximum deviation from the midline. Let's see the vertical distance from midline to peak. The midline is the average of max and min. So midline y = (3 + (-1))/2 = 1. Then the peak is at y=3, so distance from midline (1) to peak (3) is 2. Trough is at y=-1, distance from midline (1) to trough (-1) is 2. So amplitude is 2? Wait, but wait the graph: let's count the grid. The y-axis has grid lines at 1, 2, 3 and -1. The peak is at y=3, trough at y=-1, midline at y=1. So amplitude is 3 - 1 = 2. Yes, that's correct.

Answer:

2