3.5: Graphs of Functions

Determining if a function is even or odd:

In order to determine whether a function is odd even, or neither you must plug in (-x) for the variable x.

Examples:

f(x) = –3x² + 4

f(–x) = –3(–x)² + 4

= –3(x²) + 4

= –3x² + 4

Because the final expression is the same as the original the function is even.

f(x) = 2x³ – 4x

f(–x) = 2(–x)³ – 4(–x)

= 2(–x³) + 4x= –2x³ + 4x

Because the final expression is the opposite of the original the function is odd.

f(x) = 2x³ – 3x² – 4x + 4

f(–x) = 2(–x)³ – 3(–x)² – 4(–x) + 4

= 2(–x³) – 3(x²) + 4x + 4
= –2x³ –3x² + 4x + 4

Because the final expression is neither the same nor opposite the function is neither odd or even.

Types of Function Graphs:

y=f(x)-c - > Vertical Change

y=f(x-c) - > Horizontal Change

y=f(x)*c - > Vertical Stretch (no affects on x-intercepts)

y=f(x*c) - > Horizontal Stretch (no affect on y-intercepts)

This graph illustrates the graph y=|x|

Absolute value function

This graph illustrates the graph y=[x]

Greatest integer function

This graph illustrates a constant function

This graph illustrates y=mx+b

Linear function

This graph illustrates a quadratic function

This graph represents y=x³

Cubic function

This graph represents y= √x

Square root function

3.4 Definition of a Function

Before you go on to read this chapter, you should understand the most

PROFOUNDLY IMPORTANT thing that Mr. Wilhelm will tell us. Here it is: f(x)=y

The textbook almost always uses f(x), but have no fear.

These ideas of fear you may have rest in the fiery pits of hell for they will do you no good.

They will confuse you and tell you that math isn't your friend. And that is the biggest lie ever told.

Math is your friend.

Now that we have gotten past that obstacle, we can talk about the subject of the chapter. Functions.

The book definition of a function is pretty complicated and mean. So here's a simplified one.

Function- every x corresponds to exactly one y.

Simple. But we have to not be fooled by the evildoings of correspondences that aren't functions.

For instance, f(x) = x² is a function because every x has one y.

ex. 4=2², 4=-2²

But y²=x is not a function. it can be simplified to y=. It has 2 y's for every x.

ex. 2=√4 and -2=√4. Unacceptable.

2 cannot equal bicycle and motorcycle.

In a function, the possible values of x is called the domain.

The possible values of y is the range. Fun.

In the mysterious land of math, there is a magical game called the vertical line test.

Here's how it goes. If no two points are on any vertical line on the graph, it's a function.

If two or more points are on a vertical line........ NOT A FUNCTION

This is a function with domain [-2π, 2π] and range [-1, 1].

So I think we all learned a little something today. Yep.

Sections 3.1-3.3

Sections 3.1-3.3 Rectangular Coordinate Systems, Graphs of Equations and Lines

3.1

Section 3.1 deals with several important ideas. One of these important ideas is the distance formula. the distance formula is

this can be used to find the distance between any two points.

ex.

Another important formula is the midpoint formula. The midpoint formula is used to find the midpoint of a line.

ex.

3.2

Section 3.2 discusses graphs of equations. One way to create a graph is by plotting points.

ex.

Finding x and y intercepts.

y=x+4

set y=0 to find x intercept and x=0 to find the y intercept.

0=x+4

x=-4

x intercept is (-4,0)

y=0+4

y=4

y intercept is (0,4)

equation of a circle

the equation for a circle is (x-h)^2 + (y-c)^2=r^2 where (h,k) is the center of the circle and r is the radius.

how to find the center and radius of a circle.

in order to find the center and radius of a circle you must complete the square.

ex.

x^2+y^2- 4x + 6y=3

to find the center and radius you have to complete the square

(x^2 + 4x + _)+ (y^2 + 6y +_)=3+_+_

add (4/2)^2 and (6/2)^2. Don't forget to add to the other side

Then just simplify and you get

(x-2)^2 + (y+3)^2 =16

3.3

The line is a very important part of all math and should be understood thuroughly.

The slope of a line and being able to find it is also very important.

To find the slope you use the equation

ex.

This brings us to our next equation which is slope intercept form. This is expressed as

y=mx=b

where m stands for slope and b stands for the y intercept.

How to find equations for vedrticle or horizontal lines.

Find a line parallel to the x axis and a line parallel to the y axis that passes through the point (-5,6)

for the one parallel to the x axis the equation is x=-5

and for the one that is parallel to the y axis y=6

How to find the equation of a line that is parallel to a given line

ex.

6x+3y=4

3y=-6x+4

y=-2x+3

We have now found the slope , which is nessicary in order to find a parallel line.

Now plug in the other points, and the known slope into point slope form.

y+7=-2x+10

y=-2x+3

For finding a perpendicular line just do the same thing except take the opposit reciprical for the slope. For example in this case the slope in the second equation would be 1/2.

Matthew Silbergleit

Section 2.2- Applied Problems

Due to mediocre timing, mediocre technical issues and mediocre-ness in general, this blog post is a few weeks late, however, dont let that stop you from expanding your knowledge of applied problems!

First off, there are a few helpful steps to solving applied problems "easily". such guidelines are as follows.

1. Read the problem over several times carefully and identify the facts that are given, and the value that you hope to find.

2. Assign a letter to represent the unknown quantity.

3. If appropriate, draw and label a picture.

4. List all of the known facts and a relationships involving the unknown quantity. (relationships are simply equations with written words on one or both sides of the equal sign.)

5. Formulate and equation that states the relationship from step 4.

6. solve the equation from step 5.

7. check the solution to step 6 against the origional given problem.

Simple Example: Jack had 10 apples, and then he gave 4 of the apples to jane. how many apples remain?

1. we hope to find the number of apples

2. X will represent the number of remaining apples

3. no picture is necessary

4. 10 apples - 4 apples = X apples

5. 10 - 4 = X

6. 10 - 4 = 6

7. Jack had 10 apples, then he gave 4 to Jane. Jack was left with 6 apples

Obviously, this problem was so easy most of you probably lost brain cells just from reading it. and if you did have trouble with the math involved in this problem, please speak with your teacher immidialely. Next, we'll try a little more complicated problem involving perentages and discounts.

Example: Jimmy Dean went to the store to buy breakfast treats. he bought a sandwich for $2.59, a bagel for $3.19, and a pack of breakfast sausage. he paid a total of $7.02 after a 20% discount on all three items. how much was a pre-sale cost of the sausage?

1. we hope to find the cost of the breakfast sausages, and the total pre-sale cost of the food.

2. let X represent the price of the sausages, and Y represent the pre-sale food cost.

3. no picture is necessary.

4. $2.59 + $3.19 + price of sausage = total presale price

total presale price * o.20 = $7.02

5. $2.59 + $3.19 + X = Y

Y * 0.80 = $7.02

6. Y = $5.78 + X

0.80($5.78 + X) = $7.02

$4.62 + 0.80X = $7.02

0.80X = $2.4

X = $3.00

7. $3.00 + $3.19 + $2.59 = $8.78

$8.78 * 0.80 = $7.02

As you can see that problem was almost as easy as the first, just with more complicated numbers, but as long as you follow the steps for applied problems, you too can enjoy math! I hope this post helped you all with the homework you did weeks ago, and i wish you all a very mediocre day!

2.6 Inequalities

Hi Everyone!

Today in class we learned about inequalities, which are statements that express that two quantities or expressions are not equal by using the signs...

(less than)

(greater than)

(less than or equal to)

(greater than or equal to)

Solving inequalities is similar to solving equations except for a few differences:

When you divide by a negative number, you must switch the inequalities sign:

Ex.

You must also switch the sign if you take the reciprocal of both expressions:

Ex.

Also, when you're solving an inequality, you can't square both sides or take the square root of both sides.

After you solve an inequality you must put your answer into interval notation:

You use parenthesis to represent less than/greater than, and use brackets to express less than or equal to/greater than or equal to:

Ex.

Interval Notation:

Ex.

Interval Notation:

Ex.

Interval Notation: (- infinity, 5)U(8, infinity)

*U=Union

You can also show your answers to inequalities as graphs, like the ones below:

open dot: less than/greater than

closed dot: less than or equal to/greater than or equal to

So those are basically the only differences between inequalities and equations so...Let's solve some inequalities!

Ex.

divide each side of the inequality by 5

Interval Notation:

Ex.

One way to solve an inequality like this is to split it into two inequalities:

and

solve both inequalities by subtracting 7 from each side and you get...

and

Interval Notation: (-infinity, -6) U (-3, infinity)

A faster way to solve this inequality is by solving both inequalities at the same time:

subtract 7 from the whole inequality...

Interval Notation: (-infinity, -6)U(-3, infinity)

Ex.

First, multiply the whole inequality by 2

Then, divide the inequality by 3

Interval Notation:

Ex.

You know that the left side of the inequality has to be greater than zero, so in order for it to be greater than zero, the ratio must be positive. So, the denominator has to be positive to make the inequality true. Therefore....

subtract 4 from each side

Interval Notation: (-4, infinity)

Absolute Value Inequalities:

Solving absolute value inequalities is basically the same as solving absolute value equations.

Ex.

split the inequality into two inequalities: one with x-3 and another with -(x-3)

and

then, solve the inequalities in the same way you would solve a normal inequality...

and

Interval Notation: (1,5)

So, that's it. I hope this is helpful.

-Olivia