(1) graph the following function. identify amplitude, period, phase shift, vertical shift. show at least two…

Explanation:

Step1: Recall general form of sinusoidal function

The general form of a sinusoidal function is $y = A\sin(Bx - C)+D$ or $y = A\cos(Bx - C)+D$. For $y=\frac{1}{2}\sin(\pi x)+1$:

• Amplitude $|A|$. Here $A = \frac{1}{2}$, so amplitude $=\frac{1}{2}$.
• Period $T=\frac{2\pi}{|B|}$. Since $B = \pi$, $T=\frac{2\pi}{\pi}=2$.
• Phase - shift is $\frac{C}{B}$. Here $C = 0$, so phase - shift is $0$.
• Vertical shift is $D = 1$. The domain of $y=\frac{1}{2}\sin(\pi x)+1$ is all real numbers, $(-\infty,\infty)$ because the sine function is defined for all real $x$. The range is $[1 - \frac{1}{2},1+\frac{1}{2}]=[\frac{1}{2},\frac{3}{2}]$ since the range of $\sin(\pi x)$ is $[- 1,1]$.

Step2: For $y = 2-4\cos(3x)$

Rewrite it as $y=-4\cos(3x)+2$.

• Amplitude $|A| = 4$.
• Period $T=\frac{2\pi}{|B|}=\frac{2\pi}{3}$.
• Phase - shift: $C = 0$, so phase - shift is $0$.
• Vertical shift $D = 2$. The domain is $(-\infty,\infty)$. The range is $[2 - 4,2 + 4]=[-2,6]$ as the range of $\cos(3x)$ is $[-1,1]$.

Step3: For $y=\cos(4x+\frac{\pi}{2})-1$

Rewrite as $y=\cos(4x-\left(-\frac{\pi}{2}\right))-1$.

• Amplitude $|A| = 1$.
• Period $T=\frac{2\pi}{|B|}=\frac{2\pi}{4}=\frac{\pi}{2}$.
• Phase - shift $\frac{C}{B}=\frac{-\frac{\pi}{2}}{4}=-\frac{\pi}{8}$.
• Vertical shift $D=-1$. The domain is $(-\infty,\infty)$. The range is $[-1 - 1,-1 + 1]=[-2,0]$.

Step4: For $y =-\sin(\frac{\pi x}{3}-\pi)$

• Amplitude $|A| = 1$.
• Period $T=\frac{2\pi}{|B|}=\frac{2\pi}{\frac{\pi}{3}}=6$.
• Phase - shift $\frac{C}{B}=\frac{\pi}{\frac{\pi}{3}} = 3$.
• Vertical shift $D = 0$. The domain is $(-\infty,\infty)$. The range is $[-1,1]$.

Step5: For $y = 3\csc(\frac{\pi}{2}x)$

The general form of the cosecant function is $y = A\csc(Bx - C)+D$. Here $A = 3$, $B=\frac{\pi}{2}$, $C = 0$, $D = 0$. The period of $y=\csc(Bx)$ is $\frac{2\pi}{|B|}$, so the period of $y = 3\csc(\frac{\pi}{2}x)$ is $\frac{2\pi}{\frac{\pi}{2}}=4$. The domain: $\csc(\frac{\pi}{2}x)=\frac{1}{\sin(\frac{\pi}{2}x)}$, so $\sin(\frac{\pi}{2}x)\neq0$. Then $\frac{\pi}{2}x\neq k\pi,k\in\mathbb{Z}$, $x\neq2k,k\in\mathbb{Z}$. The domain is ${x\in\mathbb{R}:x\neq2k,k\in\mathbb{Z}}$. The range is $(-\infty,-3]\cup[3,\infty)$.

Step6: For $y=\sec(2x)-3$

The general form of the secant function is $y = A\sec(Bx - C)+D$. Here $A = 1$, $B = 2$, $C = 0$, $D=-3$. The period of $y=\sec(Bx)$ is $\frac{2\pi}{|B|}$, so the period of $y=\sec(2x)-3$ is $\pi$. The domain: $\sec(2x)=\frac{1}{\cos(2x)}$, so $\cos(2x)\neq0$. Then $2x\neq(2k + 1)\frac{\pi}{2},k\in\mathbb{Z}$, $x\neq(2k + 1)\frac{\pi}{4},k\in\mathbb{Z}$. The domain is ${x\in\mathbb{R}:x\neq(2k + 1)\frac{\pi}{4},k\in\mathbb{Z}}$. The range is $(-\infty,-4]\cup[-2,\infty)$.

Step7: For $y=\cot(\frac{x}{2}+\frac{\pi}{4})$

Rewrite as $y=\cot(\frac{x}{2}-\left(-\frac{\pi}{4}\right))$. The period of $y = \cot(Bx)$ is $\frac{\pi}{|B|}$, so the period of $y=\cot(\frac{x}{2}+\frac{\pi}{4})$ is $\frac{\pi}{\frac{1}{2}}=2\pi$. The domain: $\cot(\frac{x}{2}+\frac{\pi}{4})=\frac{\cos(\frac{x}{2}+\frac{\pi}{4})}{\sin(\frac{x}{2}+\frac{\pi}{4})}$, so $\sin(\frac{x}{2}+\frac{\pi}{4})\neq0$. Then $\frac{x}{2}+\frac{\pi}{4}\neq k\pi,k\in\mathbb{Z}$, $x\neq2k\pi-\frac{\pi}{2},k\in\mathbb{Z}$. The domain is ${x\in\mathbb{R}:x\neq2k\pi-\frac{\pi}{2},k\in\mathbb{Z}}$. The range is $(-\infty,\infty)$.

Step8: For $y=\frac{1}{2}\tan(2x-\pi)+3$

The period of $y=\tan(Bx)$ is $\frac{\pi}{|B|}$, so the period of $y=\frac{1}{2}\tan(2x - \pi)+3$ is $\frac{\pi}{2}$. The domain: $\tan(2x-\pi)=\frac{\sin(2x - \pi)}{\cos(2x - \pi)}$, so $\cos(2x-\pi)\neq0$. Then $2x-\pi\neq(2k + 1)\frac{\pi}{2},k\in\mathbb{Z}$, $x\neq k\pi+\frac{3\pi}{4},k\in\mathbb{Z}$. The domain is ${x\in\mathbb{R}:x\neq k\pi+\frac{3\pi}{4},k\in\mathbb{Z}}$. The range is $(-\infty,\infty)$.

Answer:

(a) Amplitude: $\frac{1}{2}$, Period: $2$, Phase - shift: $0$, Vertical shift: $1$, Domain: $(-\infty,\infty)$, Range: $[\frac{1}{2},\frac{3}{2}]$ (b) Amplitude: $4$, Period: $\frac{2\pi}{3}$, Phase - shift: $0$, Vertical shift: $2$, Domain: $(-\infty,\infty)$, Range: $[-2,6]$ (c) Amplitude: $1$, Period: $\frac{\pi}{2}$, Phase - shift: $-\frac{\pi}{8}$, Vertical shift: $-1$, Domain: $(-\infty,\infty)$, Range: $[-2,0]$ (d) Amplitude: $1$, Period: $6$, Phase - shift: $3$, Vertical shift: $0$, Domain: $(-\infty,\infty)$, Range: $[-1,1]$ (e) Amplitude: N/A (cosecant has no amplitude in the traditional sense), Period: $4$, Phase - shift: $0$, Vertical shift: $0$, Domain: ${x\in\mathbb{R}:x\neq2k,k\in\mathbb{Z}}$, Range: $(-\infty,-3]\cup[3,\infty)$ (f) Amplitude: N/A (secant has no amplitude in the traditional sense), Period: $\pi$, Phase - shift: $0$, Vertical shift: $-3$, Domain: ${x\in\mathbb{R}:x\neq(2k + 1)\frac{\pi}{4},k\in\mathbb{Z}}$, Range: $(-\infty,-4]\cup[-2,\infty)$ (g) Amplitude: N/A (cotangent has no amplitude in the traditional sense), Period: $2\pi$, Phase - shift: $-\frac{\pi}{4}$, Vertical shift: $0$, Domain: ${x\in\mathbb{R}:x\neq2k\pi-\frac{\pi}{2},k\in\mathbb{Z}}$, Range: $(-\infty,\infty)$ (h) Amplitude: N/A (tangent has no amplitude in the traditional sense), Period: $\frac{\pi}{2}$, Phase - shift: $\frac{\pi}{2}$, Vertical shift: $3$, Domain: ${x\in\mathbb{R}:x\neq k\pi+\frac{3\pi}{4},k\in\mathbb{Z}}$, Range: $(-\infty,\infty)$