4.5 Graphs of Sine and Cosine Functions

Basic Sine and Cosine Waves

A sine wave is the graph of a sine function.  

y=sin x

Consider the table of values:

 The graph of y=sin x completes on cycle from [0,2π].  This makes the period of y=sin x 2π.

The domain of y=sin x is [-∞,∞].

The range is [-1,1].  This makes the amplitude of y=sin x 1.

A cosine wave is the graph of a cosine function. 

y=cos x

Consider the table of values:

The graph of y=cos x completes one cycle from [0,2π].  This makes the period of y=cos x 2π.

The domain of y=cos x is [-∞,∞]. 

The range is [-1,1].  This makes the amplitude of y=cos x 1.  

Transformations of Sine and Cosine Functions 

Consider the functions y = a sin[b(x-c)] + d and y = a cos[b(x-c)] + d where a, b, c, and d are all constants.  These constants affect the graphs of the parent functions y=sin x and y=cos x in the following ways:

a = vertical stretch factor

b = horizontal stretch factor

c = horizontal shift (phase shift)

d = vertical shift (midline shift)

Amplitude and Period of Sine and Cosine Waves

The amplitude of y=sin x and y=cos x represents half the distance between the maximum and minimum values of a function and is given by

Amplitude = |a| 

The period of y=sin x and y=cos x is the time it takes to complete one cycle of the unit circle and is given by

Period = 2π / |b|

Even and Odd Functions

Sine is an odd function:

A function f is odd if, for each x in the domain of f,

f(-x) = -f(x)

To prove this,

sin(-π/2) = -sin(π/2)

          -1 = -(1)

          -1 = -1

Also, the graph of y=sin x is symmetric about the origin.  

Cosine is an even function.

A function f is even if, for each x in the domain of f,

f(-x) = f(x)

To prove this,

cos(-π) = cos(π)

        -1 = -1

Also, the graph of y=cos x is symmetric about the y-axis.  

3.2 Logarithmic Functions and Their Graphs

The logarithmic function f with base a is denoted as:
    f(x)=log_ax
where a>0, a≠1, and x is a real number.

The logarithmic function f(x)=log_ax is the inverse of the exponential function f(x)=a^x.

 These are the graphs for f(x)=log₁₀x (in blue) and f(x)=10^x (in red).

Notice that that the logarithmic graph is a transformation of the exponential graph.

       THE KEY TO EVERYTHING

With a logarithmic function, y=log_ax implies that a^y=x.

For example:
log₂4=2
implies that
2²=4.

When evaluating logarithmic functions, it is important to keep this in mind.

For example:
   log₅125=x
(We can solve for x by remembering The Key to Everything.)
   5^x=125
(125 is equal to 5³.)
   5^x=5³
(If 5 raised to the x power is equal to 5 raised to the 3^rd power, then...)
   x=3
(...x is equal to 3.)

       THE PROPERTIES OF LOGARITHMS

• log_a1=0 because a⁰=1
• log_aa=1 because a¹=a
• log_aa^x=x because a^x=a^x
• If log_ax=log_ay, then x=y.

The natural logarithmic function, similar to the natural exponential function, uses the number e as a base.
   For example: f(x)=log_ex
The natural logarithmic function is also rewritten in this way:
f(x)=ln x where ln denotes log_e.

The common logarithmic function is a logarithm with the base 10.
   For example: f(x)=log₁₀x
The common logarithmic function is also rewritten in this way:
f(x)=log x where log denotes log₁₀.

       GRAPHS OF LOGARITHMIC FUNCTIONS

Recall that the graph of the logarithmic function f(x)=log_ax is the inverse of the exponential function f(x)=a^x.
The graph of a logarithmic function is its corresponding exponential graph reflected across the line y=x.

These are the graphs for f(x)=log₁₀x (in blue), f(x)=10^x (in red), and f(x)=x (in orange).

In order to graph a logarithmic function, you should first graph its corresponding exponential function. Because they are inverses, you can find the corresponding exponential function by switching the x and y values.



Additionally, because they are inverses, the domain of one function is the range of the other, and the range of one function is the domain of the other.

   For example:
The domain (the possible x-values) of the function f(x)=log x is (0, ∞).
Therefore, the range (the possible y-values) of the function f(x)=10^x is (0, ∞).
The domain of the function f(x)=10^x is (-∞, ∞).
Therefore, the range of the function f(x)=log x is (-∞, ∞).

Graphs of Rational Functions

Chapter 2.7 - Graphing Rational Functions

this chapter contains:
-how to analyze and sketch the graphs of rational functions
-how to decide if graphs have slant asymptotes and what they are

Guidelines for Graphing Rational Functions

1. Evaluate f(0) to find the y-intercept and then plot it on the graph.
2. Set the numerator equal to zero -> N(x) = 0. The real solutions represent the x-intercepts and should also be plotted on the graph. 
3. Set the denominator equal to zero -> D(x) = 0. The real solutions represent the vertical asymptotes. They can be plotted on the graph as dashed vertical lines.
4. Find the horizontal asymptote and sketch it on the graph as a dashed horizontal line.
5. Using the end-behavior, fill in the graph with smooth, curved lines.

Example: 

1. f(0) = -3/2 
2. 3 = 0 -> No real solutions, so no x-intercepts
3. 0 = x - 2 -> x = 2 is a vertical asymptote.
4. y = 0 is the horizontal asymptote because the degree of x in the numerator is less than the degree of x in the denominator.
5. Since no asymptotes are multiplicities, the parts of the graph are opposites. The left piece has to be below y = 0 in order to touch the y-intercept. The opposite in the right would the be above      y = 0. 

                         

Slant Asymptotes

If the degree of the numerator is exactly one more than the degree of the denominator, the graph has a slant (or oblique) asymptote.

For example: has a slant asymptote.

To find the asymptote, use long division.

The slant asymptote portion of the answer is everything but the remainder over the divider. This leaves the line  as the slant asymptote