3.1 Exponential Functions and their Graphs

Why hello again blog post. It hasn't been long enough.

Friends, faculty, family members and random internet travelers: today I am here to tell you about exponential functions and their graphs.

Exponential functions can be written in the form

 

y = a * b^(x+c) + d

or you may see

 

f(x)= e^x

e is a constant that is an irrational constant around 2.7183, while x is a variable.

So, let's look at some graphs

 

This is actually a giraffe. But it's a common mistake. If that's actually what you were searching for, here's a video you might like:
http://www.youtube.com/watch?v=VDhNutbXpFE

Majestic, aren't they?
But really when graphing an exponential function it's important to identify:

•Asymptotes
 Start from the asymptote being y=0. Then it is important to look at two variables; c and d. In the exponential equation stated above c and d are both capable of shifting the graph up and down the graph.

•Intercepts
The x and y intercepts are found the same way as in any function. Plug in 0 for the y value to find the x-intercept and plug in 0 for the x value to find the y-intercept.

•Whether the graph is increasing or decreasing
If x doesn't have a negative coeffecient then the graph will increase. If the graph looks something like y=a*b^(-x) then it will decrease. The graph will also decrease if  0<b<1. Assuming a is postive. If a is negative then will the graph will reflect over the x-axis, reversing everything I just said.

It's also important to note how much the graph has been stretched by the value of b. As b gets larger the graph will be horizontally compressed and the opposite as it gets smaller. And the how much the graph has been shifted needs to be taken into account.

Some very basic examples:

A practical use of exponential functions is in compound interest.

Compound interest can be expressed in the formula

                 

A=P(1+r/n)^(nt)

 

A is the balance of the account, P is principal, t is the number of years, r is an annual interest rate and n is the number of times per year that the account compounds.

 

 

The book lickers must be stopped.

Operations Research is the place to be next year.

Peter's spheres are no match for mine.

Go frustums

 

Andrew

2.2 Polynomial Functions of Higher Degree

Graphs, Intercepts, and Extrema

Graphs

Here are some examples of graphs of polynomials, we can use these to help illustrate the relationship between degree of polynomials and the number of intercepts or extrema.

Zero Degree Polynomial

First Degree Polynomial

Second Degree Polynomial

Third Degree Polynomial

Fourth Degree Polynomial

Fifth Degree Polynomial

Intercepts

To find the number of intercepts in a polynomial function one can use the graph of the function or use the degree of the polynomial. When we look at the graphs the number of intercepts is either equal or less than the degree of the polynomial (when there are zeros of 2 or 3 multiplicity).

Number of intercepts (i) of a polynomial of degree (n) can be found with  i ≤ n

Extrema

To find the number of maxima and minima of a polynomial one can use to the graph of the function or use the relationship between degree and number of extrema.

Number of extrema (e) of a polynomial of degree (n) be found with e = n-1

Limit Notation

Limit Notation is used to describe End Behavior, or how a function behaves as it approaches ∞ or -∞.

Example for f(x)=-5x^2+23 

Left End:                                                          Right End:

lim f(x)=-∞                                                       lim f(x)=∞

x -> ∞                                                               x -> ∞

"The limit of f(x) as x approaches infinity"

In order to predict end behavior one can calculate values for a massive positive number then calculate the value of a very large negative number. Also one can use the characteristics of odd and even graphs to predict end behavior.

When the highest degree is Even: limits will be identical

When the highest degree is Odd: limits will be opposite

Steps:

1. Find term with largest degree

2. Look at the coefficient in front of term

If the coefficient is negative and the degree is even, both limits are -∞

If the coefficient is negative and the degree is odd, the right limit is -∞ and the left limit is ∞

If the coefficient is positive and the degree is even, both limits are ∞

If the coefficient is positive and the degree is odd, the right limit is ∞ and the left limit is -∞

That is all.

Tommy McLeod