## Precalculus (6th Edition) Blitzer

When the graph of $y=\sin 2x$ is shifted one unit upward, then the graph of $y=\sin 2x+1$ is obtained. For the standard function in the form $y=A\sin \left( Bx+C \right)$ \begin{align} & \text{Amplitude=}\left| A \right| \\ & \text{Period=}\frac{2\pi }{B} \\ & \text{Phase}\,\text{Shift}=\frac{C}{B} \end{align} $\text{Quarter-period}=\frac{2\pi }{4}$ So, Amplitude is 0 The period is given below: \begin{align} & \text{Period = }\frac{2\pi }{B} \\ & =\frac{2\pi }{2} \\ & =\pi \end{align} Phase shift is 0 The quarter-period is as follows: $\text{Quarter-period}=\frac{\pi }{4}$ Now, add quarter periods starting from $x=0$ to generate x-values for the key points. The x-value for the first key point is as follows: $x=0$ ] And the x-value for the second key point is: \begin{align} & x=0+\frac{\pi }{4} \\ & =\frac{\pi }{4} \end{align} And the x-value for the third key point is: \begin{align} & x=\frac{\pi }{4}+\frac{\pi }{4} \\ & =\frac{\pi }{2} \end{align} And the x-value for the fourth key point is: \begin{align} & x=\frac{\pi }{2}+\frac{\pi }{4} \\ & =\frac{3\pi }{4} \end{align} And the x-value for the fifth key point is: \begin{align} & x=\frac{3\pi }{4}+\frac{\pi }{4} \\ & =\pi \end{align}