Single Variable Calculus, Chapter 1, 1.3, Section 1.3, Problem 8

(a). What is the relationship of the graph of $y= \sin 2 x$ to the graph of $y=\sin x$? Use transformations to sketch the graph of $y=\sin 2 x$.




To obtain the graph $y=\sin 2 x$ from $y=\sin x$, we need to compress the graph of $y=\sin x$ by a factor of 2 horizontally as shown below:




(b). What is the relationship of the graph of $y= 1+\sqrt{x}$ to the graph of $y= \sqrt{x}$? Use transformations to sketch the graph of $y=1+\sqrt{x}$.





To obtain the graph of $y= 1+\sqrt{x}$ from $y=\sqrt{x}$, we need to shift the graph of $y=\sqrt{x}$ 1 unit upward as shown.