Find a b and c for the function f(x) = a sin(bx – c) such that the graph of f matches the graph on the right.?

Ans. a=-2, b=4/5 & c=pi/10

Hint:

f(x)=asin(bx-c)

f '(x)=abcos(bx-c)

f "(x)=-a(b^2)sin(bx-c)

At(-pi/2, 2),

asin(-bpi/2-c)=2

cos(-bpi/2-c)=0

-a(b^2)sin(-bpi/2-c)<0

=>

you can find out a,b & c.

Check:

f(-pi/2)=

-2sin[4(-pi/2)/5-pi/10]=

-2sin[-2pi/5-pi/10]=

-2sin[-5pi/10]=

-2sin[-pi/2]=

2

The amplitude, in the world's smallest image, would be 2. There are 3 periods in what looks like a 12pi wide domain, so the period is 4pi. That means that b = 1/2

f(x) = 2 * sin((1/2) * x - c)

f(-pi/2) = 2

2 = 2 * sin((1/2) * (-pi/2) - c)

1 = sin(-pi/4 - c)

pi/2 = -pi/4 - c

3pi/4 = -c

c = -3pi/4

f(x) = 2 * sin((1/2) * x - (-3pi/4))

f(x) = 2 * sin((1/2) * x + (3pi/4))