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).

If f is an odd function, then it’s graph is symmetric with respect to the origin and f(-x)= -f(x).

f x = cos x

f x = sin x

Evaluate f(-x) and determine if each function is even, odd or neither.

22

2

2sin 31. ( ) cos 3. ( )

2

2. ( ) sin 4. ( ) sin

xf x x x f x

x

f x x x f x x x

Recall: The Intermediate Value Theorem (IVT) p.206 in Pre-Calc Text

0

0

If a and b are real numbers with and if is continuous on the interval , ,

then takes on every value between ( ) and ( ). In other words, if is between

( ) and ( ), then = ( ) for

a b f a b

f f a f b y

f a f b y f c

some number in , .

In particular, if ( ) and ( ) have opposite signs (i.e., one is negative and the

other is positive), then ( ) 0 for some number in , .

Note: The Intermediate Value Theorem i

c a b

f a f b

f c c a b

s an

existence theorem. It indicates whether at least

one c exists, but does not give a method for

finding c.

Making Sense of the IVTThink 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

21. ( ) sin(2 ) on ,

6 3

22. ( ) 2cos ( )sin on ,

3 3

f x x

f x x x

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 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.

2( ) 2sin 1f x x

Example 3: Answer the following questions about on [0, 4π]. What are the zeros of f ?

Do a number line sign study for f .

Identify the intervals for which f < 0.

2 1 12 2( ) 2sin sin( )f x x x

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 Tmrrw!!