2nd Hour Honors Pre-Calculus Winter 2013: Honors' PreCalc

Sunday, January 27, 2013

3.1 Exponential Functions and Their Graphs

Hello all!
Over the chapters and through the tests, to Chapter Three we go...

To start out, let's be like Mr. Wilhelm and slip in some definitions. What are exponential and logarithmic functions? They are otherwise known as transcendental functions; they can't be expressed in terms of algebra.

An Exponential Function is denoted by;
This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

BUT:
0<a

1 and x is any real number.

Graphing
For example purposes, the two functions we'll use are This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

and

Note that as the base number gets larger, the graph increases more rapidly. (Sorry about the scale for the y-axis being different.)
So let's go over the basics for this find of function;
-Domain: This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

-Range:

-Intercept: This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

-Increases
-The x-axis is the horizontal asymptote
-Continuous

But what happens if you do something crazy, like put -x as the exponent? Well, what a great question you have...
Let's use the same equations but make -x the exponent.
This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

and

Woah! The graph goes the OPPOSITE way! Crazy! But the same general idea holds true; the larger the base, the more rapidly the graph decreases.
Just like regular algebra, a function with a negative exponent can be written as a fraction instead. For example; This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

can also be written as This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

.
The basics for this kind of function are...
-Domain: This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

-Range:

-Decreasing
-The x-axis is the horizontal aymptote
-Continuous

What to do next...hmm...transforming the parent function sounds fun!
This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

So many possibilities, let's go alphabetically.
When you change a, the graph has a vertical stretch(or compress depending on whether 1>a or a>1).
Changing b horizontally stretches/compresses the graph(unless you make it negative, in which case it flips over the x-axis).
If you were to change c, it would horizontally shift the graph left/right.
Lastly, the changing of d would result in a vertical shift of the graph.

Natural Base
We can't forget about our friend, e. This little guy is known as the natural base, and it's function is This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

. This e is in fact an irrational number that is approximated to 2.71828... but luckily, your calculator should have this as a key so you don't have to type all of the number in. Just be sure you realize that e isn't a variable, e has value just like 1 and 2, just not as exact.You plug e into your calculator just like any other number.
As far as graphing functions of e, all of the ideas we already went over apply. Just instead of using a constant, put e in as the base. Everything else is the exact same.

Compound Interest
This is fairly straightforward, as long as you know what the formulas and variables are. For n compoundings per year, the equation is This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

.
-A stands for balance in an account
-P is principal
-r means annual interest rate(expressed as a decimal)
-n is the number of compoundings per year
-t stands for the amount of time in years

For continuous compounding, the equation is This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

.
The variables are all the same as in the previous formula, just don't forget that e is an actual number, NOT a variable.

I'm pretty sure that covers the section. If my post doesn't answer all of your questions, ask a buddy, be a book-licker or ask the all knowing, Mr.Wilhelm.

Thursday, November 29, 2012

Polynomial Inequalities

Hello, again.

That was cute.
Anyways, I'm here to make music and explain polynomial inequalities, and I'm all out of music. Seriously, I'm not in band this tri. It's awful. I haven't had 5 academic classes in a row for an extended period of time since 4th grade. But enough about the real world. Let's talk of polynomial inequalities.

One of the first things we did to start class this year was define some things:

Variable-A letter or symbol that represents a certain quantity.

Numeric Expression-Any set of numbers and/or variables and/or operations

Equation-Two expressions with one side equaling another.

Solution-A number or set of numbers (interval, perhaps?) that a variable may be to make an equation true.

So what is an inequality? I propose that an Inequality is an expression that shows the relative value of two other expressions. Examples:
X>Y

X is greater than Y

How about we add an operation or two?

2X>Y+9

How about we have fun with this?

6(X^2+12X+32)>Y/5

Oh my, it seems we have stumbled upon quite a brain buster. Well, you're supposed to treat inequalities the same as equations when it comes to solving, so let's do that. We'll try to find the zeros, so let's set Y=0 for simplicity's sake.

5*6(X^2+12X+32)>0

We'll, we can cancel some numbers here. We do, after all, have the zero product property, don't we? Let's cancel the 5*6

X^2+12X+32>0
Wait, this totally looks like it factors.

(X+4)(X+8)>0

Well, we know from the zero product property that either X+4>0 or X+8 is greater than zero, right? so X>-4 or X>-8.

Wait a minute...

If X>-4 or X>-8, doesn't that mean that X could be any number above -4 or -8, not just zero? So they could multiply into something other than 0? There are infinitely more things they could multiply into! Therefore, THE ZERO PRODUCT PROPERTY DOES NOT APPLY TO INEQUALITIES!!!!!!

What it can do for us, however, is give us test values by setting the in equality into an equation.
That in mind, let's change X=-4 and X=-8 into test values.

We have essentially three segments of a number line here to plot the solutions of this inequality.
We have everything below -8, everything between -8 and -4, and everything above negative 4. So, what we need to do is pick some arbitrary points on the number line and plug them in to our in equality that is set for 0. All that matters, since 0 is our reference, is whether the number is positive or negative. Let's do -9, -5, and 0

-9^2+12(-9)+32=+

-5^2+12(-5)+32=-

0+0+32=+

So, we see that the values of X are positive (>0) at all numbers below -8 and above -4.

So our number line would end up looking something like this...

And the graph would look something like this...

And that is how we deal with polynomial inequalities. It's the same with cubes and all other degrees of polynomials: get zero on one side of the inequality and find your test values, then test them.

A brief refresher is always nice. Honors' Precalc is tough. Chin up, eyes foreward. We're all in this together. Let's make this one heck of a learning experience.
-Shane McPartlin