MATH 120 Lecture 26: How do I get sinusoidal functions into the “standard” form?
How do I get sinusoidal functions into the “standard” form?
When confronted with a sinusoidal function, it is helpful to have it (or get it into) the form
f(x) = Asin 2π
B(x−C)+D
since we are able to read off the parameters A, B, C, and D; these parameters determine
the shape and location of the graph of function. We are always able to put sinusoidal
functions into this form with A > 0, and B > 0.
Frequently, sinusoidal functions do not appear to us in this form, but we can rewrite in
that form. The following are three examples of how this goes.
1. f(x) = sin(3x+ 1)
We factor out the 3 from the expression 3x+ 1 to get
f(x) = sin 3x+1
3.
We want 2π
B= 3, so B=2
3π. Thus,
f(x) = sin 2π
2
3πx+1
3!.
And we can see that A= 1, B =2
3π, C =−
1
3, D = 0.
2. f(x) = sin(2 −7x)
We can factor the −7:
f(x) = sin −7−
2
7+x.
In order to end up with B > 0we have to do something about that −7. We can use
the following identity:
sin(−x) = sin(x+π)for all x.
This gives us
f(x) = sin −7−
2
7+x= sin 7−
2
7+x+π= sin 7−
2
7+x+π
7
= sin 7x−2
7
−
π
7= sin 2π
2
7πx−2
7
−
π
7!.
So A= 1,B=2
7π,C=2−π
7,D= 0.
3. f(x) = −
3
5sin(5 −x)
We can use the identity
−sin(x) = sin(−x)for all x
to get a positive Avalue:
f(x) = 3
5sin(−(5 −x)) = 3
5sin(x−5) = 3
5sin 2π
2π(x−5).
So A=3
5, B = 2π, C = 5, D = 0.
Document Summary
When confronted with a sinusoidal function, it is helpful to have it (or get it into) the form f (x) = a sin(cid:18)2 . B (x c)(cid:19) + d since we are able to read off the parameters a, b, c, and d; these parameters determine the shape and location of the graph of function. We are always able to put sinusoidal functions into this form with a > 0, and b > 0. Frequently, sinusoidal functions do not appear to us in this form, but we can rewrite in that form. The following are three examples of how this goes: f (x) = sin(3x + 1) We factor out the 3 from the expression 3x + 1 to get. And we can see that a = 1, b = 2. 3 , d = 0: f (x) = sin(2 7x) We can factor the 7: f (x) = sin(cid:18) 7(cid:18) .
