## Precalculus (6th Edition) Blitzer Consider the provided function, $y=2\sin \left( 2x+\frac{\pi }{2} \right)$ The general sine function equation looks as follows: $y=A\sin \left( Bx+c \right)$ From both equations above: $A=2,\text{ }B=2\text{ and c}=\frac{\pi }{2}$ Therefore, amplitude is: $\left| A \right|=\left| 2 \right|=2$ \begin{align} & \text{Period=}\frac{2\pi }{B} \\ & =\frac{2\pi }{2} \\ & =\pi \end{align} If c is a positive real number, the graph of $f\left( cx \right)$ is the graph of $y=f\left( x \right)$ stretched horizontally by $c$ units. If h is a positive real number, the graph of $hf\left( x \right)$ is the graph of $y=f\left( x \right)$ stretched vertically by $h$ units. If c is a positive real number, then the graph of $f\left( x+c \right)$ is the graph of $y=f\left( x \right)$ shifted to the left c units. Now, the provided function is: $y=2\sin \left( 2x+\frac{\pi }{2} \right)$ The graph of above function can be seen as transformations of the parent function $y=\sin \left( x \right)$, Stretch the graph of $y=\sin \left( x \right)$ by 2 units horizontally and stretch the graph vertically by 2 units. And also shift the graph to left by $\frac{\pi }{2}$.