How do you graph y=4sin(1/2x)+2?

Start with the basic $y = \sin x$ and then adjust

Explanation:

First let's look at the basic sin graph:

graph{sinx [-6.25, 6.25, -5, 10]}

And now let's make our adjustments.

Let's first look at the 4 and the 2. These change the min and max y values. The 4 expands the graph from the usual min = -1 and max = 1 by a factor of 4 to min = -4 and max = 4. The 2 lifts the graph up by 2, and so that makes min = -2 and max = 6.

That looks like this:

graph{4sin(x)+2 [-6.25, 6.25, -5, 10]}

With me so far?

Ok - so the $\frac{1}{2}$ - that will take an x value, say $x = \pi$ and have the the function evaluate it as if it were $\frac{\pi}{2}$. This expands the function along the x-axis by a factor of 2.

This is what the adjustment for the $\frac{1}{2}$ looks like without the changes to the y values (the 4 and 2):

graph{sin((1/2)x) [-6.25, 6.25, -5, 10]}

Putting it all together, we get:

graph{4sin((1/2)x)+2 [-6.25, 6.25, -5, 10]}

midline of the graph is at $y = 2$
min y value $- 2$
max y value $6$