Answer:
y=-5cos(pi/8(x+2))-21
Step-by-step explanation:
y=acos(b(x-c))+d
Let's find the max and then the min first. We are aiming to find the amplitude, a, which is the half the distance from the max to the min.
Max is y=-16
Min is y=-26
a=(-16--26)/2=-8--13=5
a could also tell us if y=cos(x) has been reflected about the x-axis if it is -5, which it looks like it has considering cos(0)=1 not -1.
I'm looking at the point (-2,-26) which is the lowest point of our curve and comparing it to our high point of (0,1) on y=cos(x).
So far we have, y=-5cos(b(x-c))+d.
Let's look at d, vertical shift. The average of our max and min is (-26+-16)/2=-13-8=-21.
So our vertical shift is d=-21.
So now equation is y=-5cos(b(x-c))+-21.
Now since we are using (-2,-26) as our (0,1) point on y=cos(x) then c=-2.
So now equation is y=-5cos(b(x--2))+-21
Now we are going to plug in a point haven't mentioned to find b. So looking at graph I will use (6,-16).
-16=-5cos(b(6--2))+-21
Add 21 on both sides:
5=-5cos(b(6--2))
Divide both sides by -5:
-1=cos(b(6--2))
Simplify inside:
-1=cos(b8)
Multiplication is commutative:
-1=cos(8b)
8b=pi * cos(pi)=-1
b=pi/8
The equation is
y=-5cos(pi/8(x--2))+-21
Simplify:
y=-5cos(pi/8(x+2))-21
