the function $g(x)$ represents $f(x)=9cos(x - \frac{pi}{2})+3$ after translating $\frac{pi}{6}$ units left…

the function $g(x)$ represents $f(x)=9cos(x - \frac{pi}{2})+3$ after translating $\frac{pi}{6}$ units left and 4 units up. which equation represents $g(x)$?\n$g(x)=9cos(x - \frac{pi}{3})-1$\n$g(x)=9cos(x - \frac{pi}{3})+7$\n$g(x)=9cos(x - \frac{2pi}{3})-1$\n$g(x)=9cos(x - \frac{2pi}{3})+7$

the function $g(x)$ represents $f(x)=9cos(x - \frac{pi}{2})+3$ after translating $\frac{pi}{6}$ units left and 4 units up. which equation represents $g(x)$?\n$g(x)=9cos(x - \frac{pi}{3})-1$\n$g(x)=9cos(x - \frac{pi}{3})+7$\n$g(x)=9cos(x - \frac{2pi}{3})-1$\n$g(x)=9cos(x - \frac{2pi}{3})+7$

Answer

Explanation:

Step1: Apply horizontal - shift rule

For a function (y = f(x)), a left - shift of (h) units gives (y = f(x + h)). Here, (f(x)=9\cos(x-\frac{\pi}{2}) + 3) and (h=\frac{\pi}{6}). So the function after the left - shift is (y = 9\cos((x+\frac{\pi}{6})-\frac{\pi}{2})+3=9\cos(x-\frac{\pi}{3})+3).

Step2: Apply vertical - shift rule

For a function (y = f(x)), a vertical shift of (k) units up gives (y = f(x)+k). Here, (k = 4). So the function (g(x)) after the vertical shift is (g(x)=9\cos(x - \frac{\pi}{3})+3 + 4=9\cos(x-\frac{\pi}{3})+7).

Answer:

(g(x)=9\cos(x-\frac{\pi}{3})+7) (the second option)