Absolute Value Functions

So we have worked on absolute values and we've talked about functions.
WHICH WAS A HOOT!

But now we should talk about another thing. The "Chapter" is called Absolute Value Functions but it would be more accurate to call it graphing absolute values.

Let's look at a build of a Absolute Value equations

4|x+3|-5 = y

Now let's replace that with variables.

a|x-h|+k = y

a = 4

-h = 3

k = -5

This is the basic build of an Absolute Value Function. This is also very important in graphing. HEre is the graphing of the equation.

If you look at the vertex you will see that it's coordinates are (-3,-5). In Absolute Value Equations, The vertex is always (-h,k) and the slope is a.

Functions and their Graphs

A relations is a mapping of input and output values. The set of input values is the domain and the set of output values is the range.

So what is a function? Well a function is a relation that has an outcome that is predictable. What does that mean?

Well look at these three situations

Every time you give 75 cents, you get 1 coke. 

If you give 75 cents you will get either no cokes, 1 coke or 2 cokes.

No mater how much you give, you get a coke.

Which ones are actually predictable? 

That's right, the 1st and 3rd one! Why?

Because no matter what, you know what your out put will be. For the first one you know the outcome will always come the one input. As for the third one, you always know what is going to come out.

So what is a good way to test this? Well there are two ways.

One way is to graph the points and put lines through each point. If a line passes two points, it is not a function.

Another is to separate the input and outputs and match them, to see if one input has two outcomes.

Solving Absolute Value Equations and Inequalities

What? A lesson already?

Yes
Any way, you may wonder, what an absolute value is. Well the Absolute Value of a number is the distance from 0 to that number on a number line. ITs written like this: | x |


If c > 0 then | ax+b | = c. If c < 0 then it does not work. You can never have a negative absolute value. 
To solve an absolute value problem you have to do 2 separate equations

For | 2x-5 | = 9

2x-5 = 9 and 2x-5 = -9

then solve

2x = 14  2x = -4

x = 7 and x = -2

Now practice with some problems.

| x+7 | = 24

| x+3 | = 58

| 7x+9 | = 78 

Now can you figure out how to do this?

4| 5x-9| +23 = 76

It's not that complicated! All you have to do is solve for the absolute value. See?:

4| 5x-9| +23 = 76

4| 5x+9 | = 53

| 5x+9 | = 53/4

Oi vey

Now, you should always now that when solving an Absolute Value, you may get what is known as an Extraneus Solution. Take | x-6 | = -3+2x Why don't you solve this one

When you are finally done you should have x = 3 and x = -3. But if you check your work you will find that -3 does not fit! That means it is an Extraneus Solution. Make sure to check your work.

Lastly we have greater thens and less thens (>, <)

There is an easy way to figure this out.

When you get the two answers you either put OR or AND

You put and when the absolute is less thAND

You put or when the absolute is greatOR

See you tomorrow!

Welcome!

Welcome to this math blog. This is made so that math is easier to learn. I am going to be putting up lessons of chapters/sections of math, like algebra 2. I will be using memes to help get the idea across.