Linear Functions by Dave Sweet

Linear Functions - by Dave Sweet



Q - What is a function?

A - A function is a relationship or expression involving one or more variables.

Q - What is a linear function?

A - A linear function is a function where the input and output values increase or decrease at a constant rate, or slope. On a graph, a linear function is a straight line.

Q - What does a linear function look like?

A - A graph, table, or equation:

GRAPH

TABLE

EQUATION

Q - What is the equation for a linear function?

A - Dependent variable/output = slope/rate of change x independent variable/input + y-intercept OR

OR

y = mx + b


Q - What is slope?

A - Slope is the constant rate of change represented by m in the linear equation y = mx + b. The slope is calculated by rise/run or output/input, or (y2 - y1)/(x2 - x1).

GRAPH



TABLE

m = 3/1

EQUATION








Q - What is the y-intercept

A - The y-intercept is the point at which a line intersects the y-axis on a graph. This is the value of y when x = 0. 

GRAPH
















TABLE







EQUATION



To find b, set x = 0, calculate m and isolate b so that b = y/mx

Sit down and Shut UP

Dr. Max B Sherman
Pre-Calculus
Harvard University
4/11/2020

Hello class, I'll be your professor for this semester. Let's get a few things out of the way first:
1) Under no circumstances will I allow any technology in the classroom. This will be the only 90 minutes of your day that your not connected, so enjoy it instead of dreading it.
2) Even if you don't plan to become a Mathematican like me, pre-calculus will help you throughout your miserable lives, so pay attention
3) Your grade is not based on your test scores, but on your active particpation and willingness to learn in class. The grades are only there because Harvard requires them.

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Domain & Range

Today we will be covering the domain and range of functions and graphs. The first thing to learn is the definition of both the domain and range

The Domain: This is all the values that are inputed into the function, or found on the graph. In most cases, these are referred to as the X-values.

The Range: These are all of the outputted values. Most commonly these are considered the Y-values of a function.

R- All real numbers, this is used to to set the parameters for where the domain and range start and stop.
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Now that we've covered the basic definitions of these concepts, it is important to comprehend them in a real life analogy.

Imagine the total population of your school, this is the equivalent of R, all real numbers.

Now consider within your class who your friends are, and whom you dislike. We will consider your friends as the domain, and those you dislike as your range. We say that out of all the students are the school, (R), you are only friends with Matt, Alex, Reggie and Scotty. This is the domain and we will illustrate it as Domain (Matt, Alex, Reggie, Scotty) The same can be done for the range. These intervals help us determine who your friends with and who you dislike, just like in a graph, they help us determine which values are included, and which are excluded from the function.
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This is our first example of domain and range. you can clearly see it illustrated in this graph. However, this is only a representation of domina and range. Next class we will learn how to properly write it in set builder and interval notation.

We will countinue with this concept next week. Until then don't get in any trouble with the Robocops.

Let's study Linear function !!

Prof. Fumi:
Hi everyone. I would like to introduce to you guys about Linear function today. Does  anyone know about Linear function is?

Students:
Linear function is a  function that it has the same rates of slope.

Prof. Fumi:
perfect! The characteristics of linear function have the same rates of change as well as the line is a straight line. The formula of a linear function is following y=ax+b. a is the slope and b is y-intercept. Slope can calculate in following: m=y1-y0/x1-x0. Y-interest is the number which cross the y-axis.

Let's do some exercise!  Here is the question,

Let's write a formula of this function.
First, is it linear function or not?

Student:
It is a linear function, because the slope is a straight line.

Prof. Fumi
Yes, but can you explain more specifically how to define it is linear?

Student:
 We can define whether the rate of slope is constant or not.  So pick up two points in this case (0,4) and (2, 0).  So, the line will be (0-4)/(2-0)=-4/2=-2 so the slope is -2. let's take other two points, (1, 2) and (3, -2) the slope will be (-2-2)/(3-1)=- -4/2= -2. These four points have same rate of slope therefore, it is linear function

Prof. Fumi
Good job!!!  Then, the slope is -2, so the formula will be y=-2x+b. Form the graph above, it can be easily recognazie the y-intercept is 4, but lets asume you only provide two points (0,4) and (2, 0). What will do?

Students: we can put   (0,4) into the equation. So it will be 4 =-2*0+b. b=4. y-intercept is 4 in this case.

Prof. Fumi
Yes, you got it!! So the equation will be y= -2x+4
This is the basic concepts of Linear function.

That's all for today!!

Graphing the Slope Lecture

Hello class, I am Professor Abdulla.
Today I will be teaching you all how to graph the slope of a function in a fun and easy way.

Also referred to as rise over run, you must refer to the graph as positive, negative, zero, or undefined. Follow this story about Sam to learn how and why linear graphs are the way they are.

The following is a mountain, a desert, and a cliff. Sam decided to go on a hiking trip and is walking up the mountain.

There are two important rules to remember about the way Sam goes.
1. Sam can only travel from left to right.
2. The only item Sam has with him is a parachute.

While Sam is traveling up the mountain, does he have a positive or negative rise?
The answer is positive, because he is traveling UP the mountain.
Does he have a positive or negative run?
Also positive, because he is traveling to the left.

So once you have the rise and the run, put them in the equation used to find slope: rise over run. Since both are positive and a positive divided by a positive is a positive, that means the slope is positive.

While Sam is traveling down the mountain, does he have a positive or negative rise?
The answer is positive, because he is traveling DOWN the mountain.
Does he have a positive or negative run?
Positive, because he is still traveling to the left.
Note that the steepness of the mountain does not change the 'negative' property of the line, and the slope is now negative.

As he travels east he crosses the desert, hot and thirsty.

His rise is zero because he is going neither up nor down, and he run is positive because he is still going left. Therefore the slope is zero because zero divided by a positive number is still zero. 

Sam approaches the edge of the cliff and uses the parachute to go straight up. Therefore the rise is positive and the run is zero. This makes the slope undefined because a number divided by zero makes it undefined. 

Now you can figure out how to graph a slope using the rise over run, and figure out if they are positive, negative, zero or undefined by figuring out if it is going up or down the mountain, going across the desert or up the cliff! 

The Number "e"

Good Morning Class,

I am Professor Rhea and I will be teaching you all a little something about the number "e"

Now you might be wondering how the letter "e" can also be a number...this is possible because of the brilliant mind of Leonard Euler a Swiss mathematician. 

He found this using the following proof "(1+1/m)^m" and what this means is that as "m" reaches ∞ that (1+1/m)^m converges to "e" and when you plug m into the equation, the outcome stays at roughly 2.71

So "e" roughly equals 2.71

Or...

2.7182818284590452353602874713527 (and more ...) if you want to be exact!

Now because (1+1/m)^m approaches "e", any positive base "b" can be written as a power of "e" 

n (1 + 1/n)n 1 2.00000 2 2.25000 5 2.48832 10 2.59374 100 2.70481 1,000 2.71692 10,000 2.71815 100,000 2.71827

For example: (b = e^k) 

Because of this an exponential function f(t)=ab^t can be re-written in terms of "e" as f(t) = ae^kt

f(t) = ae^kt is used in Continous Compounding:

Where...
a = initial value
e = 2.71
k = the interest rate
t = the amount of time the interest rate is compounded over

That is it for our lecture today and if you have any questions you can visit me in my office hours or email me!

Have a good day,

Professor Rhea



Source: http://www.mathsisfun.com/numbers/e-eulers-number.html 

Logarithmic Functions by professor Atef

Logarithms are the "opposite" of exponents, just as subtraction is the opposite of addition and division is the opposite of multiplication. Logs "undo" exponentials. Technically speaking, logs are the inverses of exponentials.
In practical terms, I have found it useful to think of logs in terms of The Relationship:

On the left-hand side above is the exponential statement "y = b^x". On the right-hand side above, "log_b(y) = x" is the equivalent logarithmic statement, which is pronounced "log-base-b of y equals x"; The value of the subscripted "b" is "the base of the logarithm", just as b is the base in the exponential expression "b^x". And, just as the base b in an exponential is always positive and not equal to 1, so also the base b for a logarithm is always positive and not equal to 1. Whatever is inside the logarithm is called the "argument" of the log. Note that the base in both the exponential equation and the log equation (above) is "b", but that the x and y switch sides when you switch between the two equations.

If you can remember this relationship (that whatever had been the argument of the log becomes the "equals" and whatever had been the "equals" becomes the exponent in the exponential, and vice versa), then you shouldn't have too much trouble with logarithms.

By the way: If you noticed that I switched the variables between the two boxes displaying "The Relationship", you've got a sharp eye. I did that on purpose, to stress that the point is not the variables themselves, but how they move.

• Convert "6³ = 216" to the equivalent logarithmic expression.

To convert, the base (that is, the 6)remains the same, but the 3 and the 216 switch sides. This gives me:

log₆(216) = 3  

• Convert "log₄(1024) = 5" to the equivalent exponential expression.

To convert, the base (that is, the 4) remains the same, but the 1024 and the 5 switch sides. This gives me:

4⁵ = 1024

Slope and Rate of Change by Dr. David Varela