Question

What is the period and midline?

Answer 1

`\bf \qquad \qquad \qquad \qquad \textit{function transformations} \\ \quad \\ % function transformations for trigonometric functions \begin{array}{rllll} % left side templates f(x)=&{{ A}}cos({{ B}}x+{{ C}})+{{ D}} \\ \quad \\ \end{array}\qquad`





`\bf \begin{array}{llll} \bullet \textit{function period}\\ \qquad \frac{2\pi }{{{ B}}}\ for\ cos(\theta),\ sin(\theta),\ sec(\theta),\ csc(\theta)\\ \qquad \frac{\pi }{{{ B}}}\ for\ tan(\theta),\ cot(\theta) \end{array}`

so if you notice yours
`\bf \begin{array}{llll} 3.2cos&\left( (5)/(3)\theta \right)+&6.1\\ &\ \uparrow&\uparrow \\ &B&D \end{array}`

now.. normally the function
`\bf 3.2cos&\left( (5)/(3)\theta \right)`
has a D value of 0, or no vertical shift, and the amplitude and the period simply make the wave taller and thinner, but the midline is still the x-axis

now, with D = 6.1, that moves the midline up vertically that much

now.. the period, well, B = 5/3, normal period of cosine is
`2\pi`
so, the new period will be
`\bf \cfrac{2\pi }{B}\implies \cfrac{2\pi }{(5)/(3)}\implies \cfrac{6\pi }{5}`

notice the picture below
the vertical shift by the D component, or 6.1, moved the midline to y = 6.1 :)