Graphing sinusoidal functions (I)

Statement

Sketch the graph of the following functions:

• (a) $y=2\cos\!\left(\dfrac{x}{2}\right)-1$
• (b) $y=\dfrac{3}{2}\sin(2x)+\dfrac{1}{2}$

Solution

For $y=A\,\text{func}(kx)+q$: amplitude $|A|$, period $T=\dfrac{2\pi}{|k|}$, midline $y=q$.

(a) $A=2$, $k=\tfrac12 \Rightarrow T=\dfrac{2\pi}{1/2}=4\pi\approx12{.}57$, midline $y=-1$; values between $-3$ and $1$. A cosine starting at the maximum: at $x=0,\ y=1$; minimum $y=-3$ after half a period ($x=2\pi$).

(b) $A=\tfrac32$, $k=2 \Rightarrow T=\dfrac{2\pi}{2}=\pi\approx3{.}14$, midline $y=\tfrac12$; values between $-1$ and $2$. A sine starting on the midline going up: at $x=0,\ y=\tfrac12$; maximum $y=2$ at $x=\tfrac{\pi}{4}$.

$\boxed{\text{(a) }A=2,\ T=4\pi,\ \text{midline }y=-1;\quad \text{(b) }A=\tfrac32,\ T=\pi,\ \text{midline }y=\tfrac12}$