Graphing sinusoidal functions (II)

Statement

Sketch the graph of the following functions:

• (a) $y=\dfrac{2}{3}\cos\!\left(\dfrac{4x}{3}\right)-2$
• (b) $y=\dfrac{1}{2}\sin\!\left(\dfrac{3x}{2}\right)+\dfrac{3}{2}$

Solution

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

(a) $A=\tfrac23$, $k=\tfrac43 \Rightarrow T=\dfrac{2\pi}{4/3}=\dfrac{3\pi}{2}\approx4{.}71$, midline $y=-2$; values between $-\tfrac83$ and $-\tfrac43$. A cosine starting at the maximum: at $x=0,\ y=-\tfrac43$.

(b) $A=\tfrac12$, $k=\tfrac32 \Rightarrow T=\dfrac{2\pi}{3/2}=\dfrac{4\pi}{3}\approx4{.}19$, midline $y=\tfrac32$; values between $1$ and $2$. A sine starting on the midline going up: at $x=0,\ y=\tfrac32$.

$\boxed{\text{(a) }A=\tfrac23,\ T=\tfrac{3\pi}{2},\ \text{midline }y=-2;\quad \text{(b) }A=\tfrac12,\ T=\tfrac{4\pi}{3},\ \text{midline }y=\tfrac32}$