consider the graphs of $f(x)=5sin(2x - \frac{pi}{3})-9$ and $g(x)=5sin(2x - \frac{pi}{8})-9$. which feature…

Explanation:

Step1: Recall general form of sine - function

The general form of a sine - function is $y = A\sin(Bx - C)+D$, where $A$ is the amplitude, $B$ is used to find the period ($T=\frac{2\pi}{|B|}$), $C$ gives the phase - shift ($\frac{C}{B}$), and $D$ is the vertical shift. For $f(x)=5\sin(2x-\frac{\pi}{3}) - 9$, we have $A_f = 5$, $B_f=2$, $C_f=\frac{\pi}{3}$, $D_f=-9$. For $g(x)=5\sin(2x-\frac{\pi}{8}) - 9$, we have $A_g = 5$, $B_g=2$, $C_g=\frac{\pi}{8}$, $D_g=-9$.

Step2: Analyze period

The period of $f(x)$ is $T_f=\frac{2\pi}{|B_f|}=\frac{2\pi}{2}=\pi$. The period of $g(x)$ is $T_g=\frac{2\pi}{|B_g|}=\frac{2\pi}{2}=\pi$. So the periods are the same.

Step3: Analyze amplitude

The amplitude of $f(x)$ is $|A_f| = 5$, and the amplitude of $g(x)$ is $|A_g| = 5$. So the amplitudes are the same.

Step4: Analyze phase - shift

The phase - shift of $f(x)$ is $\frac{C_f}{B_f}=\frac{\frac{\pi}{3}}{2}=\frac{\pi}{6}$. The phase - shift of $g(x)$ is $\frac{C_g}{B_g}=\frac{\frac{\pi}{8}}{2}=\frac{\pi}{16}$. So the phase - shifts are different.

Step5: Analyze vertical shift

The vertical shift of $f(x)$ is $D_f=-9$, and the vertical shift of $g(x)$ is $D_g=-9$. So the vertical shifts are the same.

Answer:

the phase shift