** problem 1)**
Based on the graph below, what is the period?

To solve these problems, just start at the x-axis and look for the first time that the graph returns to that 'height.' So, in this case, we're looking for the time when the graph returns to the -.5 value which is at $$ 2 \pi$$.

** problem 2**
Based on the graph below, what is the period?

Remember: Find the height of the graph at the x-axis and then look for the first time that the graph returns to that height. In this case, the answer is $$ \pi $$
or just $$ \pi $$.

Graphs generated by http://www.meta-calculator.com

** problem 3**
What is the period of the equation $$ y = -2sin(x) $$ ?

Remember: The formula for the period only cares about the coefficient, $$ \color{red}{a} $$ in front of the x. The formula for the period is the coefficient is **1** as you can see by the 'hidden' 1 :
$$ -2sin( \color{red}{1}x) $$
$
period = \frac{2 \pi}{ \color{red}{a}}
\\
period = \frac{2 \pi}{1}
\\
period = 2 \pi
$

** problem 4**
What is the period of the equation $$ y = -7cos(8x) $$ ?

Remember: The formula for the period only cares about the coefficient, $$ \color{red}{a} $$ in front of the x. The formula for the period is the coefficient is **8** :
$$ -7 cos ( \color{red}{8}x) $$
$
period = \frac{2 \pi}{ \color{red}{a}}
\\
period = \frac{2 \pi}{ 8}
\\
period = \frac{ \pi}{4}
$

** problem 5**
What is the period of the equation $$ y = 3cos(-2x) $$ ?

So, the big question here is: what do we do about the negative sign? Well, the answer is, we do not worry about the negative sign. Period tells us how long something is, and it must be a positive humber.
$$ 3cos( -\color{red}{2}x) $$
$
period = \frac{2 \pi}{ \color{red}{a}}
\\
period = \frac{2 \pi}{2}
\\
period = \pi
$