Chapter 2.1- Quadratic Functions

Helpful Videos:
Changing a Quadratic from Standard Form to Vertex From:
http://www.youtube.com/watch?v=dse3XdN9q2k
Vihart video about parabolas:
http://www.youtube.com/watch?v=v-pyuaThp-c

Chapter 1.3

Lesson 1.3- Shifting, Reflecting, and Stretching Graphs!

Graphs of Common Functions:

The six most commonly used functions in algebra are as follows:

1.  Constant: 

2) Identity:

3) Absolute Value:

4) Square Root:

5) Quadratic:

6) Cubic:

Rigid Transformations:

Rigid Transformation- type of graph transformation that does not alter the basic shape of the graph in anyway, leaving the basic shape unchanged.

Vertical and Horizontal Shifts:

The first common type of graph transformations are shifts. Shifts are rigid transformations.

Depending on the placement of a number will determine if it shifts the graph vertically or horizontally and its sign will determine which direction it will go. 

If c is a positive real number than...

 the graph is shifted c units upward

 the graph is shifted c units down

 the graph is shifted c units right

 the graph is shifted c units left    *Note: when c is inside parenthesis it shifts graphs in the opposite direction that what would be assumed. 

Reflecting Graphs:

the second common type of graph transformations are reflections. Reflections, like shifts, are also rigid transformations

When a negative sign is present in front of x or f(x), the graph will reflect over the opposite axis. Therefore...

 

represents a reflection in the x-axis

 represents a reflection in the y-axis

Nonrigid Transformations- 

Vertical Stretches and Compresses

Because vertical stretches and compresses are nonrigid transformations, they cause distortions or changes in the shape of the original graph. Vertical stretches and compresses only appear as the value in front of f(x).

 the graph is vertically stretched by multiplying all y values by c

 the graph is vertically compressed by multiplying all y factors by (1/c)

Translating Math to English

 \[x_{1}\] is x with 1 as a subscript and  \[x_{2}\] is x with 2 as a subscript

For a lot of people, math is sometimes hard to understand. For people like me, math is always hard to understand. If you're like me, and don't understand mathematics, it is probably because mathematicians devote their lives to finding ways to confuse and deceive non-mathematicians. This does not mean, however, that you cannot understand mathematics. By simplifying the language in math, you can suddenly understand much more than you previously thought you could. Here are some examples of translating math to English:

Math talk: A function f is increasing on an interval if, for any \[x_{1}\] and  \[x_{2}\] in the interval, \[x_{1}< x_{2}\] implies \[f(x_{1})< f(x_{2})\].

Wow. Sounds tricky, doesn't it? But that is only because of the way it is written. If we break it down, we can understand the meaning. First, we should say that an interval is a section of a graph.  Now let's look at " for any \[x_{1}\] and  \[x_{2}\] in the interval, \[x_{1}< x_{2}\] ". This simply means that there are two x-values on a graph, and one of those values lies to the left of(is less than) the other point. Now for "  \[f(x_{1})< f(x_{2})\] ".  This means that the y-value that corresponds to the x-value on the left is below(less than) the y-value that corresponds to the x-value on the right. An x and y value make a point. So one point is to the left of and below the other. In the end, we get this " A function is increasing at a certain section if, in that section, any point that is to the left of another point is also below that point."

Peace bro-

Andrew Geller

http://www.youtube.com/watch?v=v-pyuaThp-c

Difference Quotients

A difference quotient is typically thought of to be just an equation, but it is simply a more complex variation of the slope formula.

The equation for a difference quotient is: 

In class, we derived this formula.  The picture below shows a slight variation of how this formula may be derived.  The difference quotient can be used to find the slope of individual portions of graphs of polynomial or exponential functions.  As you can see, if you use the basic slope formula given the values of x and y (x,f(x)) and (x+h,f(x+h)), the simplified result is the difference quotient.

Caution:When solving difference quotient problems, never try to set anything equal to zero; it is only an expression.

Here are a few example problems of how to solve difference quotient problems given a function f(x):

And with a polynomial function:

Have a great day!!!!! Julia Wilkins

Functions and Their Graphs: 1.1 - Functions

A function is relationship involving one or more variables. Functions have an input and output. The input is the first coordinate of the ordered pair and the output is the second coordinate. In the example below, the input values are a, b, c, and d, and the output values are p, q, and r.

Note the following characteristics of the function above that will apply to other functions:    

(1) Each element in A is paired with an element of B

(2)  Some elements in B may not be paired with any element in A

(3)  Two or more elements of A may be matched to the same element of B

Also, if an element of B were to be paired with two elements of A, then the picture would not represent a function.

i.e.                 a-->p                                   a-->p

                                              b -->q                                 b-->q

                                              b-->r                                   c-->q

                                              c-->p                                  d-->r

                                    NOT A FUNCTION                 FUNCTION

Functions must pass the vertical line test:

Function Notation is just a way to represent that the equation is a function.

To evaluate a function, you must enter the input value into the output, and then solve. This next example shows the evaluation of g(x)= 3/(10-7x) for the solution of x when we have g(x)=g(2):

A Piecewise-Defined Function is simply a function in Piecewise Form, which may sound confusing, but the main idea is shown below (the lines show the solutions to the function):

Videos to help find Domain of functions:

Algebraically: http://www.youtube.com/watch?v=w81y25anEOM

Graphically: http://www.youtube.com/watch?v=ObEucyZX464

The domain is all values of the independent variable for which the function is defined.

Difference Quotient:

***http://www.mathwords.com/d/difference_quotient.htm

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