Question

Which cosine function has maximum of 2, a minimum of -2, and a period of 2pi/3?

Answer 1

i tried to google this and it came up with weird stuff about scheduling. this is worded oddly and idk how to solve it

Answer 2

Hints:
A typical cosine function can be described as follows:
\(\Large f(x)=Acos(B(x-C))+D\)
A is the amplitude, equal to (maximum-minimum)/2, it acts as a vertical scale factor.
B is relates to the period, and period equals \(2\pi/B\), works as a horizontal scale factor.
C is horizontal shift.
D is vertical shift.
In the given case, C and D are zero, because there is no shift (translation).
Normally cos(x) has a maximum of +1 and minimum of -1.
A will amplify cos(x) to the given values (vertical scale factor).
To find B, you would solve the equation:
\(\Large \frac{2\pi}{B}=\frac{2\pi}{3}\)

For more information and examples, read:
http://www.regentsprep.org/regents/math/algtrig/att7/sinusoidal.htm

Answer 3

okay, i still dont really understand though

Answer 4

well, going off of what mathmate said

Answer 5

our basic template is

Acos(B(x−C))+D

Answer 6

the maximum/minimum of the basic cos function is 1 and -1, but in this case our minimum/max is -2 and 2 instead

Answer 7

that means, we need to multiply the entire function by 2

Answer 8

what is the function

Answer 9

this Acos(B(x−C))+D

Answer 10

we go from

Acos(B(x−C))+D

to

2cos(B(x-C)) + D

Answer 11

okay

Answer 12

C is our horizontal shift, and since we don't have a horizontal shift, C = 0

Answer 13

D is vertical shift, and we don't have one here either, so D = 0

Answer 14

that means we go from

2cos(B(x-C)) + D

to

2cos(Bx)

last step is to find B

Answer 15

our equation is

2pi/B = period

Answer 16

so

since period = 2pi/3

2pi/B = 2pi/3

what does B equal?

Answer 17

3

Answer 18

great

Answer 19

so our final function is

2cos(3x)

Answer 20

that's it

Answer 21

oh okay

Answer 22

thank you