Question

name: syanie smith
n for each graph

1. y =

bonus
graph of a wave - like function with x - axis marked -2π, -π, 0, π, 2π and y - axis marked -5, 5
4.
bonus
graph of a curve - like function with x - axis marked -π, 0, π, 2π and y - axis marked 5

Explanation:

Step1: Identify base function (Graph 2)

The graph is symmetric about the y-axis, matches $\cos(x)$ shape, no reflection.

Step2: Find amplitude (Graph 2)

Max y-value is 2, so amplitude $A=2$.

Step3: Find period (Graph 2)

Distance between peaks is $2\pi$, so period $T=2\pi$, $\omega=\frac{2\pi}{T}=1$.

Step4: Check shifts (Graph 2)

No horizontal/vertical shift, $C=0$, $D=0$.

Step5: Form equation (Graph 2)

$y = A\cos(\omega x + C) + D$

Step6: Identify base function (Graph 4)

Graph is U-shaped, matches $\sec(x)$ shape, no reflection.

Step7: Find vertical stretch (Graph 4)

Minimum y-value is 2, so stretch factor $A=2$.

Step8: Find period (Graph 4)

Distance between minima is $2\pi$, so period $T=2\pi$, $\omega=\frac{2\pi}{T}=1$.

Step9: Check shifts (Graph 4)

No horizontal/vertical shift, $C=0$, $D=0$.

Step10: Form equation (Graph 4)

$y = A\sec(\omega x + C) + D$

Answer:

1. $y=2\cos(x)$
2. $y=2\sec(x)$