5. based on the provided graph:\na. whats the amplitude?\nb. whats the period?\nc. what is the midline?\nd…

Explanation:

Part a: Amplitude

Step1: Recall amplitude formula

Amplitude ( A=\frac{\text{Max}-\text{Min}}{2}). From the graph, (\text{Max} = 4) and (\text{Min}=-2). [A=\frac{4 - (-2)}{2}=\frac{4 + 2}{2}=3]

Part b: Period

Step1: Identify period from graph

The period is the length of one full cycle. Observing the graph, from (x = - 2) to (x=2) (or any two consecutive corresponding points of the same phase), the length is (4).

Part c: Midline

Step1: Use midline formula

Midline (y=\frac{\text{Max}+\text{Min}}{2}). Substitute (\text{Max} = 4) and (\text{Min}=-2). [y=\frac{4+(-2)}{2}=\frac{4 - 2}{2}=1]

Part d: Sine - equation

Step1: General form of sine function

The general form is (y = A\sin(B(x - C))+D). We know (A = 3), (D = 1), (B=\frac{2\pi}{\text{Period}}), and since (\text{Period}=4), (B=\frac{\pi}{2}). The graph is a sine - graph shifted. Using the point ((0,-2)): (-2=3\sin\left(\frac{\pi}{2}(0 - C)\right)+1) (-3 = 3\sin\left(-\frac{\pi}{2}C\right)) (\sin\left(-\frac{\pi}{2}C\right)=-1) (-\frac{\pi}{2}C=-\frac{\pi}{2}) (one - solution), so (C = 1) The equation is (y = 3\sin\left(\frac{\pi}{2}(x - 1)\right)+1=3\sin\left(\frac{\pi}{2}x-\frac{\pi}{2}\right)+1)

Part e: Cosine - equation

Step1: General form of cosine function

The general form is (y=A\cos(B(x - C))+D). We know (A = 3), (D = 1), (B=\frac{\pi}{2}). Using the point ((-2,4)): (4=3\cos\left(\frac{\pi}{2}(-2 - C)\right)+1) (3=3\cos\left(-\pi-\frac{\pi}{2}C\right)) (\cos\left(-\pi-\frac{\pi}{2}C\right)=1) (-\pi-\frac{\pi}{2}C = 2k\pi), for (k = - 1), (-\pi-\frac{\pi}{2}C=-2\pi) (\frac{\pi}{2}C=\pi), (C = 2) The equation is (y = 3\cos\left(\frac{\pi}{2}(x - 2)\right)+1=3\cos\left(\frac{\pi}{2}x-\pi\right)+1)

Answer:

a. (3) b. (4) c. (y = 1) d. (y=3\sin\left(\frac{\pi}{2}x-\frac{\pi}{2}\right)+1) e. (y = 3\cos\left(\frac{\pi}{2}x-\pi\right)+1)