Q.From the curve $y=\sin x$, draw $y=\sin|x|$ (Hint: $\sin(-x)=-\sin x$.)

Step 1. For $x\ge0$: $|x|=x$, so $\sin|x|=\sin x$ -- identical to the ordinary sine curve on the right half.

Step 2. For $x<0$: $|x|=-x$, so $\sin|x|=\sin(-x)$. Using the hint $\sin(-x)=-\sin x$ (sine is odd), we get $\sin|x|=-\sin x$ for $x<0$.

Step 3. So on the left half ($x<0$), the graph of $y=\sin|x|$ is the graph of $y=-\sin x$ -- which is exactly the MIRROR IMAGE (about the $y$-axis) of the right-half sine curve, since reflecting $\sin x$ about the $y$-axis gives $\sin(-x)=-\sin x$ for the corresponding negative $x$. …

Split by the sign of $x$, simplify $\sin|x|$ using the hint, and describe the …

• Assuming $\sin|x|$ just equals $|\sin x|$ (which would fold the graph upward instead) -- absolute value INSIDE …