Download 4.8 Number line sign studies for composite trigonometric functions

4.8 Symmetry, IVT and
Number line sign studies
for composite trig
functions
Recall the definitions of even/odd
functions:

If f is an even function, then it’s graph is
symmetric with respect to the y-axis and
f(-x)=f(x).
fx = cos x

If f is an odd function, then it’s graph is
symmetric with respect to the origin and
f(-x)= -f(x).
fx = sinx
Evaluate f(-x) and determine if each
function is even, odd or neither.
1. f ( x)  x cos x
2sin 3 x
3. f ( x)  2
x 2
2. f ( x)  x  sin x
4. f ( x)  sin x  x 2
2
Recall: The Intermediate Value Theorem
(IVT) p.206 in Pre-Calc Text
If a and b are real numbers with a  b and if f is continuous on the interval  a, b  ,
then f takes on every value between f (a) and f (b). In other words, if y0 is between
f (a ) and f (b), then y0 =f (c) for some number c in  a, b .
In particular, if f (a) and f (b) have opposite signs (i.e., one is negative and the
other is positive), then f (c)  0 for some number c in  a, b .
Note: The Intermediate Value Theorem is an
existence theorem. It indicates whether at least
one c exists, but does not give a method for
finding c.
Making Sense of the IVT
Think of the Intermediate Value Theorem as “crossing a river.” In the picture
below, if you are walking on a continuous path from f(a) to f(b), and there is a
river across your path at the horizontal line y=y0 , then you would have to cross
the river to reach your destination.
 River 
Use the Intermediate value Theorem to
determine if a zero must exist on the
interval:
  2 
1. f ( x)  sin(2 x) on  ,
 6 3 
  2 
2. f ( x)  2 cos ( x) sin x on  ,
 3 3 
2
Note: the fact that the IVT does not guarantee a zero does not mean that
one does not exist in the interval. For instance, check f(π/2) in number 2.
Example 1: Answer the following
questions about f ( x)  2sin 2 x  1 on [0, 2π].

What are the zeros of f ?

Describe the symmetry of f.

Do a number line sign study for f and use
interval notation to identify where f > 0.
Example 3: Answer the following
questions about f ( x)  2sin 2  12 x   sin( 12 x)
on [0, 4π].

What are the zeros of f ?

Do a number line sign study for f .

Identify the intervals for which f < 0.
Assignment
A4.8, Sections I, II and III to be completed
by Monday
 Test #11 will be at the end of this week
and includes Polar Equations and
Complex Numbers.


See you Tmrrw!!