10. f(x) = 5/3 cos(π/5(x + 1)) - 3/4 amplitude: period: phase shift: midline: y = 12. y = 11 sin(22π(x +…

Explanation:

Step1: Recall general form of cosine function

The general form of a cosine - function is $y = A\cos(B(x - C))+D$, where $A$ is the amplitude, $T=\frac{2\pi}{B}$ is the period, $C$ is the phase - shift, and $y = D$ is the mid - line. For the function $f(x)=\frac{5}{3}\cos(\frac{\pi}{5}(x + 1))-\frac{3}{4}$, we have:

• Amplitude: The amplitude $A$ is the absolute value of the coefficient of the cosine function. So, $A=\left|\frac{5}{3}\right|=\frac{5}{3}$.
• Period: Given $B = \frac{\pi}{5}$, using the formula $T=\frac{2\pi}{B}$, we substitute $B$ into the formula: $T=\frac{2\pi}{\frac{\pi}{5}}=10$.
• Phase Shift: The phase - shift is given by $C$. Since the function is in the form $y=\frac{5}{3}\cos(\frac{\pi}{5}(x+1))-\frac{3}{4}=\frac{5}{3}\cos(\frac{\pi}{5}(x - (- 1)))-\frac{3}{4}$, the phase - shift $C=-1$.
• Midline: The mid - line is given by $y = D$. Here, $D =-\frac{3}{4}$.

Step2: Recall general form of sine function

The general form of a sine function is $y = A\sin(B(x - C))+D$, where $A$ is the amplitude, $T=\frac{2\pi}{B}$ is the period, $C$ is the phase - shift, and $y = D$ is the mid - line. For the function $y = 11\sin(22\pi(x + 94))+23$, we have:

• Amplitude: The amplitude $A$ is the absolute value of the coefficient of the sine function. So, $A = 11$.
• Period: Given $B = 22\pi$, using the formula $T=\frac{2\pi}{B}$, we substitute $B$ into the formula: $T=\frac{2\pi}{22\pi}=\frac{1}{11}$.
• Phase Shift: Since the function is in the form $y = 11\sin(22\pi(x+94))+23=11\sin(22\pi(x - (-94)))+23$, the phase - shift $C=-94$.
• Midline: The mid - line is given by $y = D$. Here, $D = 23$.

Answer:

For $f(x)=\frac{5}{3}\cos(\frac{\pi}{5}(x + 1))-\frac{3}{4}$: Amplitude: $\frac{5}{3}$ Period: $10$ Phase Shift: $-1$ Midline: $y =-\frac{3}{4}$

For $y = 11\sin(22\pi(x + 94))+23$: Amplitude: $11$ Period: $\frac{1}{11}$ Phase Shift: $-94$ Midline: $y = 23$