CHAPTER TWOTrigonometry

NEGATIVE ANGLE IDENTITIES

sin(-θ) = - sinθcos(-θ) = cosθtan(-θ ) = - tanθcsc(-θ ) = - cscθsec(-θ) = secθcot(-θ ) = - cotθ

PERIODIC FUNCTIONS

Periodic Function- a function whose graph has a repeating pattern that continues indefinitely. The shortest repeating portion is called a cycle.

Period- The horizontal length of each cycle is called the period.

EXAMPLES OF PERIODIC FUNCTIONS

FUNCTIONSSymmetric about the y axis

Symmetric about the origin

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

8

-2-3-4-5-6-7

So for an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph.

Even functions have y-axis Symmetry

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

8

-2-3-4-5-6-7

So for an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph.

Odd functions have origin Symmetry

A function is even if f( -x) = f(x) for every number x in the domain.

So if you plug a –x into the function and you get the original function back again it is even.

125 24 xxxf Is this function even?

1251)(2)(5 2424 xxxxxfYES

xxxf 32 Is this function even?

xxxxxf 33 2)()(2NO

A function is odd if f( -x) = - f(x) for every number x in the domain.

So if you plug a –x into the function and you get the negative of the function back again (all terms change signs) it is odd.

125 24 xxxf Is this function odd?

1251)(2)(5 2424 xxxxxfNO

xxxf 32 Is this function odd?

xxxxxf 33 2)()(2YES

If a function is not even or odd we just say neither (meaning neither even nor odd)

15 3 xxf

Determine if the following functions are even, odd or neither.

1515 33 xxxf

Not the original and all terms didn’t change signs, so NEITHER.

23 24 xxxf

232)()(3 2424 xxxxxf

Got f(x) back so EVEN.

ASYMPTOTES

x

xxf

22

52

Vertical Asymptotes: Will occur at the x values that make the denominator 0.

HORIZONTAL ASYMPTOTES If the degree of the numerator is less than the degree of the

denominator. Asymptote Y=0

If the degree of the numerator is equal to the degree of the denominator. Asymptote is the horizontal line y = a/b

When the degree of the numerator is greater than the degree of the denominator there is no horizontal asymptote!!

FINDING HORIZONTAL ASYMPTOTES

If

27

533

2

x

xxxf

GRAPH OF EXAMPLE 4

The horizontal line y = 0 is the horizontal asymptote.

FINDING HORIZONTAL ASYMPTOTES EXAMPLE 5

If

975

5362

2

xx

xxxg

GRAPH OF EXAMPLE 5

The horizontal dotted line at y = 6/5 is the horizontal

asymptote.

FINDING HORIZONTAL ASYMPTOTES EXAMPLE 6

If

There are no horizontal asymptotes because the degree of the numerator is greater than the degree of the denominator.

1

9522

3

x

xxxf

GRAPH OF EXAMPLE 6