Find a formula for h(x). Give an exact expression.

Question

h is a trigonometric function of the form h(x)=a cos(bx+c)+d.

Below is the graph of h(x). The function has a maximum point at (3,6) and a minimum point at (6.5, -9).

Find a formula for h(x). Give an exact expression.

h(x)=

Answer 1

From the coordinates provided we know that from the maximum to the minimum, we use the y values to determine the height of the wave is |6|+|-9|=15 so we know that a=7.5. From this we also know that since a=7.5 and the max value is 6 then d=6-7.5=-1.5. Using the x values from the coordinates, we know that half a cycle spans 3.5 for a full period is 7. 7=2*pi/B B=2*pi/7. As this is a cosine function, the maximum of a cosine happens as x=0 and we know the new maximum is @ x=3 from this c=-3*2*pi/7

following the format:

h(x)=a * cos(2/7*pi*x +c ) + d

we have

h(x)=7.5 * cos( 2/7 * pi * x - 6/7 * pi) - 1.5

Answer 2

a= 7.5

d=-1.5

the difference in y values from max to min = 15. 15/2 = a

max cosine = 1, min cosine =-1

7.5 x 1 - 1.5 = 6

7.5x -1 -1.5 = -7.5-1.5= -9

h(x) = 7.5cos(bx+c) -1.5

now just get a period or frequency that matches the x coordinates 3 and 6.5 to get b and c

at (3,6) cosine = 1 to get a maximum value. cosine is at a max when cos(bx+c) = 1 and bx+c=0, 360, 720...

if x=3, then 3b +c=0 and c=-3b

for (6.5,-9) to be a minimum h value, then cos(6.5b+c) =-1 that happens when the angle = 180. +n360 for any integer n. 6.5b+c = 180

c= 180-6.5b

c=-3b

180-6.5b =-3b

add 6.5b to both sides to get

180= 3.5b

b = 180/3.5 = 51 1/7

c=-3b = -153 3/7

h(x) = 7.5cos(51.4x - 154.3)-1.5