Slope

Good afternoon, my name is Professor Maria and I am here to teach you about slope. A slope is the same thing as the coefficient on the x term. In math, the slope is the number that describes both the direction and the steepness of a line. The slope can tell us if the direction of a line is decreasing, increasing, horizontal or vertical.
A few things to note:

• In order to determine if the slope is increasing, the line must be going up from left to right. This means the slope is positive.
• A decreasing slope goes down from left to right. This means the slope is negative. 
• If a line contains a slope of zero, it means the line is horizontal. 

Now that we know what slope is, I am going to show you how to find slope and why it is important in mathematics. 

• To find the slope, you need to first pick two points on your line - you will be using them to divide the difference of the y-coordiante of a point by the difference of the x-coordiante. 

• Using the equation y2 - y1 / x2 - x1 we are able to find the slope of the line. The number on the top part of the faction is how many points on the line you are going to rise and the number on the bottom of the fraction is how many points you are going to run, in other words, move left or right depending if it is negative or positive. (If the number is positive you are going to move up or to the right and if it is negative you are going to move down or to the left.)

Now that we know how to find slope, lets use it in terms of linear functions. A linear equation is the equation of any straight line -  its equation is represented in terms of y = mx + b (this equation is called the slope-intercept form)

• In slope-intercept "m" is the slope and "b" is the y-intercept.
• In order to find the slope using the slope-intercept form you pick two points on the line and replace them for the "x"and the "y" of the equation, after solving for slope you would then use that answer to plug it in for "m" and solve for "b". By doing this, you are able to find the equation of the line as well as its intercept. 

• Example 1:

Find the equation of the line that has a slope of 4 and passes through the point (-1,-6)

y = mx + b

(–6) = (4)(–1) + b

–6 = –4 + b

–2 = b

Using the points given to us and the slope, we are able to find the y-intercept. 

- What if they don't give you the slope and ask you to put the equation of a line in terms of y = mx + b?

If they don't give you the slope you can find the slope by using the points on the line. 

• Example 2:

Find the equation of the line that passes through the points (-2, 4) and (1, 2). 

Now that I have the slope and two points I can find the intercept by solving for "b" by using the information I already have. (Using points (-2, 4) and slope -2/3)

y = mx + b




4 = (– ²/₃)(–2) + b





4 = ⁴/₃ + b





4 – ⁴/₃ = b
12/3 – ⁴/₃ = b





b = 8/3




...so  y = ( – ²/₃ ) x + ⁸/₃.

Slope can also be used to find the rate of change of a certain equation. 

• Example 3:

Todd had 5 gallons of gasoline in his motorbike.  After driving 100 miles, he had 3 gallons left.  The graph at the right shows Todd's situation.

a.  Find the slope of the line.

       

b.  What does this slope tell us?
Since , we know that Todd's bike is burning .02 gallons of gasoline for every mile that he travels.   The negative value of the slope tells us that the amount of gasoline in the tank is decreasing.


c.  What is Todd's mpg?

The    tells us that Todd can drive 50 miles on one gallon of gasoline (an mpg of 50 miles per gallon). 

When teaching my students about the number "e", I would first start off with the history of "e" and Leonard Euler. Euler, a mathematician, gave "e" its name based on the following proof: 

For "m", which is arbitrary, we see the pattern shown in the picture for the expression (1+ 1/m)^m. 

As "m" reaches infinity, the expression (1+1/m)^m eventually equates approximately to 2.71 consistently, otherwise known as "e". Since (1+1/m)^m approaches "e", any positive base "b" can be written as a power of "e". For example, e^k=b. As a result of this, any exponential function f(t)=ab^t can be re-written in terms of "e" as f(t)=ab^t=a(e^k)^t=ae^kt. "k" can also be known as the continuous growth rate. 

When graphed as an exponential function, "e" would fall between Q= 2^t and Q=3^t. "e" is an important concept to know because it is necessary when calculating continuous interest rates in financial situations. It also helps in any equation that involves some sort of continuous rate. 

The number "e" tess

Hi I’m Professor Richards and today I will be teaching you about the number “e”! the start off the number “e” = 2.71 and is between 2<e<3. The reason why “e” is equal to 2.71 was figured out by Leonard Euler proof. For (obituary) M we see the pattern of the expression  (1+1/m)^m.

M	(1+1/m)m
1	2
10	2.59
100	2.70
1000	2.716
10000	2.718
	“e”

 ^for (1+1/m)^m as m= (1+1/m)^m =e converges to “e”

this is because (1+1/m)^m =e any positive base “b” can be written as a power of “e” (ie, e^k=b)

·      So any expentical function f(9)=ab^t can be rewritten in terms of “e” as f(t)=ab^t=a(e^k)^t=ae^kt

·      Now “k” is called the continuous growth rate   

F(t)= ae^kt is “b” when “k” is the continuous growth rate

·      If b>1 when “k” is positive

·      If 0<b<1 then “k” is negative

An example is

Let e^k=2 and b>1

then in e^k=in2

K=in2

K²=0.69 K is the positive when b>1

An example

Let e^k=1/2 and 0<b<1

Ine^k=in1/2

K²=in1/2

K²=-0.69

K in negitice 0<b<1

Here is an example from an everyday situation

An amount of $2,340.00 is deposited in a bank paying an annual interest rate of 3.1%, compounded continuously. Find the balance after 3 years.

The Solution is:

Use the continuous compound interest formula, A = Pe rt, with P = 2340, r = 3.1/100 = 0.031, t = 3. Recall that e stands for the Napier's number (base of the natural logarithm) which is approximately 2.7183. However, one does not have to plug this value in the formula, as the calculator has a built-in key for e. Therefore, So, the balance after 3 years is approximately $2,568.06.

Rate of Change - Michael Pepe

What is rate of change?

Rate of change is one of the most deciding factors when you are constructing a function. Rate of change usually modifies the x variable in an f(x)=x style function.

The interesting part of the a rate of change is that adding a leading number to an equation is in some senses a "transformation" of the parent function y=x, which looks like this.
 
In this circumstance, the rate of change is '1', meaning if x changes 1, y also changes 1.
If we change our rate of change, it will transform the graph be changing it's slope.
an f(x) with a rate of change of 2 could look like this red function.








How can we make sense out of this very visually obvious concept in a numerical sense?
Rate of change is two dimensional, so both x and y must be taken into account.
The formula to find rate of change is Δy / Δx, y value change divided by x value change.
This can be expressed more simply as y2 - y1 / x2 - x1.

So in the parent function it comes out like this:

y2-y1 / x2 - x1   |   1-0 / 1-0     =   rate of change 1   - using the two points (0,0) and (1,1)
We can find this based on any two points on the line.


It also works for y=2x

2-0 / 1-0  = 2 rate of change from the y=2x  (0,0) and (2,1)


Rate of change is useful useful for finding equations of linear functions, as well as identifying linear and non linear functions. In a linear function, rate of change will be constant, other functions, will not have a constant rate of change.
Rate of change is unaffected by transformations like moving left right and up and down. y=2x +3 has the same rate of change as y=2x-4 (They're parallel)

Blog Post #4 - Lesson on Slope

Hello class, my name is Professor Hutkoff substituting for your beloved Professor LIttle who is out today - and I will be teaching you all about the concept of slope.

Introduction
-First off, what is a slope?

Slope is the change in Y divided by the change in X, a good way to think about it is slope equals the rise over the run of a line on a graph.

-How does one find this "slope" that you speak of?

Slope can be calculated by the formula      with "m" representing the slope, so all you would need to find the slope would be two points for the x's and y's.

Slope can also be found just by simply looking at your graph and counting the number of units for the rise and for the run (pictured below).




-Now that I have this crazy slope thing, what can I do with it?

The slope is a part of a very important equation in Algebra called the y-intercept form which is the formula of a linear equation, y=mx+b, where again, "m" is equal to the slope. Slope is vital to this formula and along with "b," which is representative of the y-intercept in the equation, one is able to successfully graph a line by using the y-intercept form.

Examples

1.) You are given the points (4,1) and (7,5). Find the slope.
Using the slope equation mentioned previously:
You would take 5 and subtract 1 equalling 4
And you would take 7 and subtract 4 equalling 3
With your end result being 4/3 for the slope

2.) For a real world scenario of how one could use slope, we'll examine the example of measuring the steepness of a roof on a building. Say you are charged with conducting repairs on the roof of a building but first you must find the slope of the roof so that you can properly repair it.
You would use a measuring device (i.e. a ruler) to find the rise and run of the roof.
If your rise was 8ft and the span of the roof was 30ft (assuming the roof is evenly divided down the middle of the building), your run would have to be 15ft.
Rise = 8 
Run = 15           therefore 8/15 would be the sloop.


And that concludes my lesson for the day class, hopefully you are leaving today satisfied with your newfound knowledge of this amazing and applicable concept of slope. Don't forget to turn in your exit slips before you leave!

Sources:
http://www.mathwarehouse.com/algebra/linear_equation/images/slope_given_2points/slope-of-a-line-graph.gif
http://math.about.com/od/algebra1help/a/Slope_Line_Graph.htm
http://www.roofhelp.com/images/roofhelp/slope.gif

My name is Professor Aburto!

Today, I will be teaching the class about functions, including linear functions

First of all, you guys will have to know the key points about functions

• Every input has an output
• Pocess the VLT (Vertical Line test)
• Functions CAN NOT have multiple outputs for one input
• A mathematical model is a function where the outputs are dependent on the input

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Example:
Function
(y) yards : output
(t) time : input
y = f(t)
                                            Peyton Manning passing yards over 5 years

Time	Yards
2009	3500
2010	4216
2011	5857
2012	6510
2013	5218

Is this example a Mathematical model? NO

Functions can be represented in (graphs, tables, formulas, words)
T = f(h)
If q is a function of t, Q= f(t)
Q: Output, dependent
f(t): Input, Independent

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Example: Regular Sea Level VS. Sea Level Depth


This function IS linear because the rate of change is CONSTANT (look below for explanation)

This is how check for linearity


(30-16) / (34-0)= 7/17, (37-30) / (51-34)=7/17, (44-37) / (68-51)=7/17

7/17 is the constant rate of change.

Rate of Change (ROC)
Is the ROC constant at every interval?

General formula for the family of linear functions
Output = intitial + ROC * input
example:
Pressure = initial (y int) + ROC * depth pressure

Slope Intercept Fomula: y = b + mx OR mx + b

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example:
We can estimate temperature by counting the number of times a snowy tree cricket chirps in 15 sec. and adding 40. T is the output, R is the input value. Evaluate in minutes (1/4 of a minute)

T= 1/4R + 40

(T is the output, 1/4 is the rate of change, R is the input, and 40 is the y int/initial value)

R (Chirp rate)	T (Temperature)
0	40
20	45
40	50
60	55
80	60
100	65
120	70
140	80

Check for linearity, use points from the table.


(45-40) / (20-0) = 1/4 , (50-45) / (40/20) = 1/4

So, this IS a linear function because the ROC is 1/4 a constant value.

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Point Slope Formula: y-y1 = m(x-x1)
example: (40,50) point from chart
y-y1 = 1/4(x-x1)
y - 50 = 1/4 (x - 40)
y = 1/4x + 40

Transformations

Today I am going to teach you about transformations. By changing the parent function of a graph, you can alter the way in which the graph appears or moves on the grid.
Let's start with the function notation for a basic quadratic graph: f(x) = x2. A function transformation takes whatever  f(x) is and then transforms it. For example, three units higher than the quadratic parent function,  F(x) = x^2, is x^2 + 3. By simply adding a 3 to the outside, you shift the parabola up 3 units along the y-axis from it's original starting location.

Lets take a look at another transformational function F(x)=(x+3)^2. With this you see the graph shifted 3 units to the left. This is because whenever a number is placed inside the function argument, it is shifted along the x-axis. So if a number is added in the argument it shift left along the x-axis, but if a number is subtracted in the argument, it is shifted to the right along the x-axis. 

The last transformation function I am going to look at is f(x)=-x^2. When you do this you simply reflect the graph on the x-axis. When we take a look at the function  G(-x)=-x^3. this graph is a reflection on the y-axis. So -f(x) is a reflection about the x-axis and g(-x) is a reflection about the y-axis.