math 131 unit 1 section 3.4 written homework\nshow your work as in class. exact value/form means; fractions…
math 131 unit 1 section 3.4 written homework\nshow your work as in class. exact value/form means; fractions (and π where appropriate), no decimals.\n1. determine the amplitude and period of each function without graphing. $y = \\frac{4}{3}\\sin(\\frac{2}{3}x)$\namplitude______ $4/3$ period______\n2. graph each function using transformations or the method of key - points. be sure to label key points and show two cycles. use the graph to determine the domain and the range of the function. $y = 4\\sin(\\frac{\\pi}{2}x)$\namp______ $4$ period______ $\\frac{2\\pi}{\\pi/2}=2$ domain______ range______ $(-\\infty,\\infty)$\n3. graph each function using transformations or the method of key - points.
Answer
Explanation:
Step1: Recall amplitude formula for $y = A\sin(Bx)$
For a sine - function of the form $y = A\sin(Bx)$, the amplitude is given by $|A|$.
Step2: Find amplitude of $y=\frac{4}{3}\sin(\frac{2}{3}x)$
Here $A = \frac{4}{3}$, so the amplitude is $\left|\frac{4}{3}\right|=\frac{4}{3}$.
Step3: Recall period formula for $y = A\sin(Bx)$
The period of the sine - function $y = A\sin(Bx)$ is given by $T=\frac{2\pi}{|B|}$.
Step4: Find period of $y=\frac{4}{3}\sin(\frac{2}{3}x)$
Here $B=\frac{2}{3}$, so $T = \frac{2\pi}{\frac{2}{3}}=2\pi\times\frac{3}{2}=3\pi$.
Step5: For $y = 4\sin(\frac{\pi}{2}x)$
The amplitude: Since $A = 4$, the amplitude is $|4| = 4$.
Step6: For $y = 4\sin(\frac{\pi}{2}x)$
The period: Since $B=\frac{\pi}{2}$, then $T=\frac{2\pi}{\frac{\pi}{2}}=2\pi\times\frac{2}{\pi}=4$. The domain of a sine function $y = A\sin(Bx)$ is all real numbers, i.e., $(-\infty,\infty)$. The range of a sine function $y = A\sin(Bx)$ is $[-|A|,|A|]$. For $y = 4\sin(\frac{\pi}{2}x)$, the range is $[-4,4]$.
Answer:
- Amplitude of $y=\frac{4}{3}\sin(\frac{2}{3}x)$ is $\frac{4}{3}$, period is $3\pi$.
- Amplitude of $y = 4\sin(\frac{\pi}{2}x)$ is $4$, period is $4$, domain is $(-\infty,\infty)$, range is $[-4,4]$.