# What cosine function represents an amplitude of 3, a period of π, no horizontal shift, and a vertical shift of?

Sep 12, 2017

In order to answer this I have assumed a vertical shift of $+ 7$

$\textcolor{red}{3 \cos \left(2 \theta\right) + 7}$

#### Explanation:

The standard cos function $\textcolor{g r e e n}{\cos \left(\gamma\right)}$ has a period of $2 \pi$

If we want a period of $\pi$ we need to replace $\gamma$ with something that will cover the domain "twice as fast" e.g. $2 \theta$.
That is $\textcolor{m a \ge n t a}{\cos \left(2 \theta\right)}$ will have a period of $\pi$.

To get an amplitude of $3$ we need to multiply all values in the Range generated by $\textcolor{m a \ge n t a}{\cos \left(2 \theta\right)}$ by $\textcolor{b r o w n}{3}$ giving
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b r o w n}{3 \cos \left(2 \theta\right)}$

There is to be no horizontal shift, so the argument for $\cos$ will not be modified by any further addition/subtraction.

In order to achieve the vertical shift (that I assumed would be $\textcolor{red}{+ 7}$ [substitute your own value]) we will need to add $\textcolor{red}{7}$ to all values in our modified range:
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{3 \cos \left(2 \theta\right) + 7}$