Showing How the Properties of a Trigonometric Function Can Transform a Graph

Show the transformation of the graph of $y = \sin x$ into the graph of $y = 2 \sin (4 x - \frac{π}{2}) + 2.$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Consider the series of graphs in Figure 2 and the way each change to the equation changes the image.

Figure 2 (a) The basic graph of $y = \sin x$ (b) Changing the amplitude from 1 to 2 generates the graph of $y = 2 \sin x .$ (c) The period of the sine function changes with the value of $B,$ such that $period = \frac{2 π}{B} .$ Here we have $B = 4,$ which translates to a period of $\frac{π}{2} .$ The graph completes one full cycle in $\frac{π}{2}$ units. (d) The graph displays a horizontal shift equal to $\frac{C}{B},$ or $\frac{\frac{π}{2}}{4} = \frac{π}{8} .$ (e) Finally, the graph is shifted vertically by the value of $D .$ In this case, the graph is shifted up by 2 units.

How did it go?