# Question #6644a

Dec 23, 2017

#### Explanation:

.

This is the graph of $f \left(x\right) = \sin x$:

As you see, it has a period of $2 \pi$ and amplitude of $1$..

This is the graph of $f \left(x\right) = \sin 2 x$:

As you can see, it has a period of $\pi$ and amplitude of $1$. You have to divide the period of the sine function by the coefficient of your angle to get the period of your new function. In your problem, your angle is $x$ and its coefficient is $2$. When you divide $2 \pi$ by $2$ you get $\pi$.

Note that the amplitudes of both functions we graphed are $1$.

This is the graph of $f \left(x\right) = 4 \sin 2 x$:

As you see, it has a period of $\pi$ and amplitude is $4$ which tells you that if you have a coefficient behind the sine function, it gets multiplied by the amplitude of a normal sine function which is $1$ and becomes the new amplitude, in this case $4$.

This is the graph of $f \left(x\right) = 1 + 4 \sin 2 x$:

The constant that gets added to the function is the $y$-shift or vertical shift of the graph. In our problem, it is $+ 1$. This means that the graph of $f \left(x\right) = 4 \sin 2 x$ moves up by $1$ unit in the $y$ direction.

If you compare $4 \sin 2 x$ with $1 + 4 \sin 2 x$, you will see that it has moved up one unit.

If you follow this process you can easily graph your trigonometric functions.