Katie's Lesson on Composition of Functions

Composition of Functions

     A composition of functions is when you combine two separate functions so that you use the output from one function for the input of the second function. (Or the range (y-values) of one function becomes the domain (x-values) of the second function. 

A composition of functions looks like this:

                                            

When reading this out loud, you say "f of g of x"

It is important to remember to go in order across from the inside out when using a composition of functions. 

When solving for f(g(x)) you first plug in your g(x) function into your f(x) function. For example, if you were given f(x)=5x and g(x)=10x+3, then your first step would be to plug g(x) into f(x).

This would look like f(g(x))=f(10x+3). Can you see how the function g(x) has taken the place of the x in the function f(x)?

The next step would be to plug the f(x) function into what you already have. So now you take your 5x and put it into the equation.

This would look like f(g(x))=5(10x+3)

                                            =50x+15

Unless you are given an x value to work with, this is as far as you can solve the problem.

You can also reverse the order of the equations and be asked to solve g(f(x))

Given the same functions as above and following the same procedure, you would find that after the first step g(f(x)) would look like g(5x)

In the second step, it would then be turned into g(f(x))=10(5x)+3

                                                                                     =50x+3

It is clear that the answers to the functions will be different depending on the order of the function.

Because of this it is extremely important to make sure you keep the functions in the correct order and plug in using the correct order. 

Let's try another example

If f(x)=4x+7 and g(x)= 3x find f(g(x)) when x=3

     f(g(x)=f(3x)
          =4(3x)+7
          =12x+7
          =12(3)+7
           =36+7

       =43          

                     
Now find g(f(x)

g(f(x))=g(4x+7)
          =3(4x+7)
         =12x+21
         =12(3)+21
         =36+21
     =57      

Now you know how to compose two functions! Good luck!

Beginning Input/Output

This link goes to my power point presentation

https://docs.google.com/presentation/d/132UT8c53JV15zX8-fDnrPjURrD-qV-rom6VB8CqaWoU/edit?usp=sharing

Nicoles be the professor

COMPLETING THE SQUARE:

 

This trick is used to factor quadratic expressions, especially when other methods (like the x-method) doesn’t work:

 

First you write your formula (ax² + bx + c) in the form ax² +bx= -c (move your c to the other side of the equation)

 

Divide b by 2, then square it (b/2)².

 

Add the (b/2)² to both sides of the equation: x² + bx + (b/2)² = -c + (b/2)².

 

Write the left side of the equation as a perfect square: [x + (b/2)]² = -c + (b/2)²

 

Let’s pretend that a=4, b=16, and c=-18

 

Move c to the right side of the equation to get 4x²+16x= 18

 

(-18 becomes positive when you move it to the right side)

 

Divide both sides of the equation by 4: x²+ 4x= 18/4.

 

Take ½ (4) which is the x term, and square it to get (4/2)²=4.

 

Add the 4 to both sides of the equation x²+ 4x +4= 18/4 + 4

 

Write the left side of the equation as (x+2)² which is a perfect square and you get that (x+2)²= 34/4

 

Then you’re done!!

 

Simple as that!

 

Well, not really, but if you do it a bunch, it eventually it will make sense. But only if you have a teacher who is patient and helpful like I do!!

Teaching Average Costs

Hello everyone, my name is Lucy Shepherd and I am here today to teach you about Average Costs. Hopefully with my help you will come to understand why average costs are important in everyday life.

 The average cost is the cost to produce an individual unit of an item based on a certain quantity of objects. Say for instance a company wants to produce 500 (q) computers. Say to build 500 computers it takes $800,000 (C(q)). The average cost to produce 1 of those 500 would be $1,600 (a(q)).

 I know this may seem like a tedious and unnecessary step, but in the long run it is really quite important. Why, you ask?

The graphs of cost functions are almost never straight. The graph of the average cost is going to be a straight graph of a linear function.

Companies that manufacture items need to keep a close eye on what they are spending per item. However, companies are not the only ones that pay attention to average cost. When you go to the supermarket and look at the price sticker on the shelf, the price per unit in the top corner is your average cost for that item. You can also find the average cost for an item in a bulk order. Say you are ordering a bulk of rubber ducks. You decide you like these rubber ducks from Oriental Trading. They cost $13.99 per 26 ducks. The average cost per duck would be $0.54 per duck. If you needed more than 26 ducks and you had a budget per duck (for some odd reason) you could then tell if the ducks were in your budget.

Thank you all for being here today. I hope this lesson gave you an insight into average costs and why they are important.

Dawson Slope Lesson

Here is a link to my lesson plan explaining slope

http://www.scribd.com/doc/220370894/Slope-Lesson-Dawson

Thanks for taking a look!

Be the Professor: blog 4

Hey guys I made this video as a lesson on product rule for derivatives.

http://www.screenchomp.com/t/41i62zb6DTS

This is my blog vid, thanks!

Be the Professor: Blog #4

Hello class! I’m Professor Jenna and today’s lesson is about domain and range. Domain and range are very important when interpreting a graph or understanding the x and y values of a function. You will need this information to analyze graphs and whatnot. 

So, what is domain? Domain is the set of input values that go into a graph or function. These points define the function. And what is range? Range is the set of all the output values of function.

For example, the surface area of a cube is the total outside area of the cube. It can be found by using the formula: A = 6x². Let us assume that you are asked to construct a cube that has a side length of at least 3 inches. In order to identify the domain, you must first identify the independent variable. Since the independent variable is the length of a side, then the domain has to describe numbers that could represent lengths of sides for this problem – the problem says the length has to be at least 3 inches, so the domain is all numbers greater than or equal to 3.

In order to identify the range, you must first identify the domain. The range is the y-value that corresponds with the chosen x-value in a function. So, to find the range, you simply find the area of a cube with a side of 3 (the smallest one you can have), and that tells me that my range has to be greater than or equal to 54.  

There are different notations for identifying domain and range. There is the set-builder notation and the interval notation:

Set-builder:

 

Interval: ( a, b)

Also, for those of you who are more visual learners, here is a graph with the domain and range specified after: 

Domain: (negative infinity, infinity) because the graph stretches out on the x-axis.

Range: (-3, infinity)

If you have any questions please let me know and have a wonderful weekend! 

Be the Professor Blog 4

Within today's lesson plan I will be explaining a key point that will follow you throughout the rest of your math careers: factoring.

Factoring for starters is actually rather simple to think about, but in practice can certain prove to be challenging when seen within complex equations. Understanding this, the easiest approach to the concept of factoring would be by starting with simple numbers, then moving on to simple equations and eventually making our way to more advanced ones.

So what exactly is a factor? It is easy to view a numbers factors as the terms that when multiplied together will give you the number that you seek. For example let us look at the number 12. The factors would be 1,12,2,6,3, and 4. Why? Because when you multiply 1*12, 2*6, and 3*4 they all will equal 12. It is also easy to view a factor as what numbers are the given number easily divisible by.

Just as single numbers can be factored, so too can variables with coefficients. In order to do this, the first step is to just find the factors of the coefficient. Knowing how to factor variables is useful for simplifying algebraic equations that the variables are among.

1. As an example the variable 12x can be written as the product of the two terms 12 and x. We then would write 12x as 3(4x) or 2(6x) ect. using which ever factors will best fit the problem that you are currently working on. 
2. You can even take the factoring one more step: look at 3(4x). You can then factory the coefficient 4 once more to produce: 3(2(2x)).

With the last two concepts it is finally time to take that knowledge of factoring both single numbers and variables with coefficients, you can move one more step to simplify algebraic equations by finding the factors that the numbers and variables in equation have in common. Usually to make the equation as simple as possible, we try to find the greatest common factor. This process is possible because of the distributive property of multiplication, which says that for any numbers a,b, and c, a(b+c)= ab+ac.

1. Let us look at an example problem. To factor the equation 12x6, first, find the greatest common factor of 12x and 6. 6 is the biggest number that divides evenly into both 12x and 6, so it can simplify the equation to 6(2x+1).

Now that you have a basic understanding of how to factor equations and numbers you should be able to tackle more and more complicated problems and they should not be as daunting. 

The only next step may be to factor a quadratic equation in order to so, it might be best to view this video and we will discuss it more in our next class: 

https://www.youtube.com/watch?v=ZQ-NRsWhOGI

Be the Professor - Blog 4

Hello Class:

Today I am going to teach you about completing the square.

What does it mean to complete the square you ask?

Well! Let me tell you!

Completing the Square is used when solving general quadratic equations.

                                                  2                        2                         2
In Algebra it looks like this:  X    + BX  + (B/2)       =   (x + b/2)


To solve a quadratic equation by completing the square.

All it takes are 




Be sure to follow along!



Divide all terms by a (the coefficient of x2).
                        

Move the number term (c/a) to the right side of the equation.



Complete the square on the right side of the equation and glance this                                                                     by adding the same value to the right side of the equation.

Take the square root of both side of the equation.


Subtract the number that remains on the left side of the equation to find x.



Example: x2 + 8x + 2


Step 1:    can be skipped because the coefficient of x2 is 1

Step 2:    x2 + 8x = -2

Step 3:    x2 + 8x + 16 = - 3 + 16  -> x2 + 8x + 16 = 13

                         2
               (x + 4)  = 14

Step 4: (x + 4) = 3.74
                  -4
Step 5:  x= .26

And thats the answer! That is how you solve for a quadratic equation by solving for the square.

Technology Anthology - Blog 3.5

I chose four websites that had games for kids about 12 years and older and that had large amounts of different games.

1.  http://www.math-play.com

A website designed to better your math skills wherever you have access to a computer.  The site has tons of different games to help you with math. The website has games split into different sections. Each category is either a level in math or a topic of math. An example would be 5th grade math or Integer games. This site is made for anyone between the ages of 7 and 20 depending on what math class you are taking.  The site has games that are appropriate up to college level calculus.  I would recommend this site to anyone looking to get some more on math and can't sit down at a computer.  The website has all sorts of games for any kind of learner or player.  The website has a good feel to it and is very organized.  If you are trying to learn more about math I definitely recommend this class. 

2. http://www.learningwave.com/abmath/

Learningwave.com holds a math game called Asburd Math.

Absurd Math is a mathematical solving questions type of game.  The player proceeds on missions in a fantasy world .  There are four different levels of the same game on the website.  This is too increase the difficulty for players as they beat some of the earlier levels.  This sight focuses on Pre-Calculus, but the four different levels include a lot more math then just Pre-Calculus.  I would not recommend this site because it does not have a lot of variety. If you get bored of the one game then there are no other games to play and all of a sudden the site is worthless.  The site needs to have more variety and then it will be a great site.

http://www.mathplayground.com/games.html

Mathplayground.com is a huge conglomerate of math games that are helpful no matter what you want to study.  The website is organized very well, which makes it easier to find what you want to do on the site.  There are games that are adding fractions while driving and other games where it asks you to fill in puzzles in order to move to the next level.  I firmly recommend this site as it is one of the far better ones out on the internet.  It has a lot of options to choose from and covers a large range of material so you will always have a new game to try if you run out of study options.

http://hotmath.com/games.html

While hotmath.com only has four games each game is intended for a separate audience.  There is Catch the Fly for middle school and up; Number Cop for middle school and up; Factortris for 9th and up; and Algebra vs the Cockroaches high school and up. The site is well organized and clean. I would recommend this site just based on the fact that all four games are fun. It also helps a lot with learning consequences in math

Be The Professor- Completing the Square

Completing the Square 

First, the goal of the mathematical concept, completing the square is to take a quadratic function, and put it in vertex form.  

Steps to take when completing the square using an example:  

x² + 16x+ 2

1) divide the middle term by two to get its half.

16/2 = 8 

2) Create a zero in the function.  This is done by taking the value found in step 1 and squaring, and then adding and subtracting your new number to the equation.

8²=64

x²+16x+64-64+2

3) Find a perfect square factor

Perfect square factor: (x²+16x+64) à (x+8)²

4) Write the problem in vertex form.

Vertex Form= f(x) = (x+8)² – 62

Vertex:  (-8, -62)

Example using a quadratic function with an "a" value of more than 1 

f(x) = 2x²+4x-5

Factor out the leading coefficient, in this case the 2, from the equation

2(x²+2x)-5

Take the half of the middle value, now a 2 instead of a 4 thanks to factoring.

2/2=1

create a zero:  f(x)= 2(x²+2x+1-1)-5

Perfect square factor: 2(x+1)² – 1(2)-5

Vertex Form: 2(x+1)² – 7

Vertex: (-1, -7)

Now that you've completed the square, you can go on to graph the function using the vertex you found!