Question

which equation matches the description? the curve has a period of 2π, an amplitude of 4, a left - phase shift of π/2, and a vertical translation down 2. y = 4 cos(2x - π/2)-2; y = 4 cos(2x + π/2)-2; y = 4 cos(x - π/2)-2; y = 4 cos(x + π/2)-2

Explanation:

Step1: Recall general cosine - function form

The general form of a cosine function is $y = A\cos(Bx - C)+D$, where $A$ is the amplitude, $B$ affects the period ($T=\frac{2\pi}{|B|}$), $C$ is the phase - shift ($\text{phase - shift}=\frac{C}{B}$), and $D$ is the vertical translation.

Step2: Analyze given parameters

We are given that $T = 2\pi$, $A = 4$, phase - shift is $\frac{\pi}{2}$ to the left (negative phase - shift), and $D=-2$. Since $T=\frac{2\pi}{|B|}=2\pi$, then $B = 1$. A left - phase shift of $\frac{\pi}{2}$ means $C=-\frac{\pi}{2}$ in the formula $y = A\cos(Bx - C)+D$.

Step3: Substitute values into the formula

Substituting $A = 4$, $B = 1$, $C=-\frac{\pi}{2}$, and $D=-2$ into $y = A\cos(Bx - C)+D$, we get $y = 4\cos(x+\frac{\pi}{2})-2$.

Answer:

$y = 4\cos(x+\frac{\pi}{2})-2$