How do you sketch y=sin (x/2)?

Jun 1, 2016

Using a 6 point summery.

Explanation:

A six point summery is identifying 6 major facts about the graph of a trig function is as followed.

$1.$ Amplitude

This is simply how high a graph will go. This is found out by looking at the coefficient in front of the trig function. In the graph of $y = \sin \left(\frac{x}{2}\right)$, there appears none but that is because it is actually $1$, and mathematicians don't write the number one out most times(if at all).

Thus, this graph will have an amplitude of $1$, which means its greatest height is 1 and lowest height is $- 1$.

$2.$ Period

The parent sine graph, $y = \sin \left(x\right)$, has a period of $2 \pi$ since it takes $2 \pi$ revolutions to complete one cycle of the graph.

To figure out the new period, simply divide $2 \pi$ by the coefficient attached to your $x$ value. In this case, the new Period is $4 \pi$, since the coefficient is one half.

$3.$ Translation of the Graph Up or Down

This is simply the question will my graph move up, down, or not at all. Again the parent sine function $y = \sin \left(x\right)$ doesn't have a translation, but if it were $y = \sin \left(x\right) + 1$ it would be moved one unit up. This graph doesn't have any, but this is part of the process

$4.$ Translation of the Graph Left or Right

This graph doesn't have any but it is also important to note. Lets say our function we must graph is $y = \sin \left(x + \pi\right)$. The entire graph would be shifted to the left one $\pi$ units. It shifts to the right because in order for the function to be $0$, it needs to have an $x$ value of $- \pi$.

$5.$ Five Number Summery

A five number summery is simply $5$ points on your graph used to map out what you will draw. I detailed it in the following paragraph.

A sine curve $\sin \left(x\right)$ starts at the origin $\left(0 , 0\right)$, has a maximum of $1$ at $x = \frac{\pi}{2}$, a zero at $x = \pi$, a minimum of $- 1$ at $x = \frac{3 \pi}{2}$, and a zero at $2 \pi$. What I just did right there is a five number summery, where I identify.

$6.$ Consider range and domain

This is huge when dealing with other trig functions. In this case the domain is all real numbers and the range is $- 1$ to $1$, written mathematically $\left[- 1 , 1\right]$.

The graph of $\sin \left(x\right)$ looks like this

graph{sinx [-10, 10, -5, 5]}

The graph of $\sin \left(\frac{x}{2}\right)$ looks like this

graph{sin(x/2) [-10, 10, -5, 5]}