Madeline R. answered • 09/27/21
Patient and Enthusiastic Tutor
To start I quickly plotted some points to find this is a cosine function. You could also write this a shifted sine function. A regular cosine function has a range of [-1,1] centered vertically on the x-axis. I would recommend plotting your points and the cos(x) function, I used Desmos online graphing calculator. This particular calculator helps you see what each change does, you can feel free to play with it a bit and get a handle on it. Compared to cos(x), our graph is stretched in both directions and shifted upwards.
In order to apply what we find, we can look at y=Acos(B(x+C))+D, where A represents the stretch and shrink along the y-axis, B represents the stretch and shrink along the x-axis, C represents a shift along the x-axis, and D represents a shift along the y-axis, and A and B are always positive.
Our graph has been stretched along the y-axis. We can see that the range is doubled, therefore, A=2.
Our graph has also been stretched along the x-axis. The point on cos(x) is at (π, -1) and ours is at (4,-1). We don't need to worry about the y value for this part. To stretch cos(x) to the right point we need B=π/4
We don't need to move our graph along the x-axis, so we can ignore C.
Our graph has been shifted up by one space, so D=1.
Our formula would look like this: 2cos(π/4x)+1
