## Trigonometry (11th Edition) Clone

$\color{blue}{y=\cos{(\frac{1}{2}x)}+1}$ or $\color{blue}{y=1+\cos{(\frac{1}{2}x)}}$
RECALL: The cosine function $y=\cos{(bx)}$ has (i) a period of $\frac{2\pi}{b}$; (ii) an amplitude of $1$; and (iii) a range of $[-1, 1]$ The given graph looks like the cosine function but has a period of $4\pi$. Thus, $4\pi=\frac{2\pi}{b} \\4\pi{b}=2\pi \\b=\frac{2\pi}{4\pi} \\b=\frac{1}{2}$ Note that the given graph has the range $[0, 2]$. This means that the graph of the function $y=\cos{(bx)}$ was shifted vertically one unit upward. Therefore, the equation of the function whose graph is given is: $\color{blue}{y=\cos{(\frac{1}{2}x)}+1}$