4.5 Graphs of Sine and Cosine Functions

The graph of a sine function is called a sine curve. The following graph represents one cycle of the sine curve and following that is the graph of the cosine function.

y= sin(x)

y= cos(x)

The domain of the sine and cosine functions is all real numbers, the range is [-1,1] and each function has a period of 2

Divide the period 2 to get 4 equal points: /2, , 3/2, and 2

Using Key Points to Sketch a Sine Curve:

y= 2 sin(x)

These key points will have twice the magnitude of the graph y= sin(x)

(0,0) (/2, 2) (, 0) (3/2, -2) (2, 0)

Amplitude and Period of Sine and Cosine Functions:

y= d + a sin(bx-c)

and

y= d + a cos(bx-c)

= amplitude of the function y= a sin(x) and y= a cos(x)

amplitude is half the distance between the maximum and minimum values of the function

ex. if y= 1/2 cos(x) the amplitude is 1/2, the maximum value is 1/2, and the minimum value is -1/2.

Divide one cycle, , into 4 equal parts to get the key points:

(0, 1/2) (/2, 0) (, -1/2) (3/2, 0) (2, 1/2)

Period=

For the rest of the translations of sine and cosine curves, consider reviewing the transformations studied in section 1.3 in our blog:

y= f(x)

y= f(x)+2 shifts up 2

y= f(x+2) shifts left 2

y= f(x-2) shifts right 2

y= 2f(x) vertically stretched

y= (2x) horizontally compressed

y= -f(x) reflects in y-axis

y= f(-x) reflects in x-axis

4.3: Identities

Identities: an identity is an equation that is always true

The following identities can be substituted in, in order to prove that one side of an equation is equal to the other.

Reciprocal Identities:

,

,

,

Quotient Identies:

Even/Odd Identities

,

,

,

cosine and secant are the only identities that are even

Pythagorean Identities:

Section 4.2

Trigonometric Functions

Triangles

The Unit Circle

In the unit circle, the x coordinate is cos and the y coordinate is sin.

Even

f(-x)=f(x)

Functions that are even: cos, sec

Odd

f(-x)=-f(x)

Functions that are even sin, csc, tan, cot.

2.7 Graphs of Rational Functions

g(x)=

First find the x and y intercept. (if any)

y-Intercept: (0,)because g(0)=

x-Intercept: There is none because when you plug in 0 for y its 30

Next we need to find the vertical and horizontal asymptote . (if any)

V.A: Just the denominator so x-2=0 x=2

H.A: Is the trickiest because its going to have 3 different scenarios

1. Powers are the same and you have to divide by leading coefficients.

2. Numerator has a larger power

3.Denominator has a larger power

1. 2. 3.

H.A= H.A= no asymptote H.A=0

So in our situation of

g(x)= when x=1,000,000,000,000 the denominator is going to be much bigger than the numerator so we call it 0.

we have everything we need to graph now.

if all the steps were done right it should come out to this

There are also situations in where we get HOLES in the graph and this occurs when we have the same factor on the numerator and denominator.

g(x)= ,, if we go to zoom decimal we can see the graph is as so

f(x)=

*rule of thumb- if one side is negative probably the other is positive