User:User0/Function dance

* Bell rings

OK, class! That was the bell and get into your seats. Hurry up now, settle down. Welcome to the first class of functions! I know you are all excited about your schedule and the new year, but there's not a lot of time to go through the all curriculum.

So, I'll be your teacher for this course and um, I see a lot of new faces in here. I think I remember you – James, I had you in grade 9, right? Yeah? Alright! Let me hand out these course outlines and we'll start right after I take attendance.

Course information

DEPARTMENT OF MATHEMATICS
Course name Function Dance	Course code MFE-4M1	Perquisites MCM-3M1
Course Description	This course introduces the mathematics of function exercise. Students will learn to the basics of functions: recognizing functions by graphs and equations, identifying the components, performing, and theoretical applications of functions. Students will also learn and use functions in practical applications that relate to occupations in mathematical careers. This course prepares the student for calculus dance.
Course outline
Term Work 70%
Unit 1	Introduction to functions
Unit 2	Advanced functions
Unit 3	Applications of functions
Summative work 30%
Cumulative project	10%
Final exam	20%
Textbook	Function 12	Publisher UnBooks

Wait while I assign you a textbook.

Unit 1

Introduction to functions

Warmup

function: A graph that does not fail the vertical line test
Cartesian graph: A graph in which is divided into 4 coordinates.
x-axis: The horizontal line of a graph in reference to (0, 0)
y-axis: The vertical line of a graph in reference to (>, <)
x-intercept: The point where the line intersects the x-axis
y-intercept: The point where the possession of the ball is gained

So, nobody coming in late? Let's start! Turn to page 4 and let's do exercises #1-4 under the “Getting Ready” section to see what you can remember.

OK, now. Can anybody identify the components of a graph? This horizontal line is the x-axis and this vertical line is the y-axis.

For a standard linear function, the form should be: $y=mx+b$ m is the slope and b is the y-intercept.

* Puts on some music

Now first we take our hands and position them horizontally on the x-axis. Make sure your arms are straight. This guys, ok, is the function f(x)=0. The slope is 0 and the y-intercept is 0. This is the first function you should know. All values of x will equal to zero. Next, make a new function with a slope of 1. Can anybody show the class?

$f(x)=x$

Good! Who knows how a slope of -1 would look like?

$f(x)=-x$

Perfect!

Now, I know you aren't used to having homework on the very first day, but the curriculum is very long and we should move at a fast pace.

More functions

domain: all possible numbers of x in a graph
range: none of the impossible numbers of f(x) in a graph
real number: numbers that are not imaginary
end behaviour: the direction of the arms of a function

Ah – it looks like we have more students. Pass out the handout and their textbook to them. So, yesterday we learned about the linear function. Today, we are going to take a quick look at the quadratic function.

$f(x)=ax^{2}+bx+c$

What are the differences between a quadratic function? Yeah, um... Janice! That's your name. Uh-huh, a quadratic function has a range which means that the values of f(x) or y is not every real number. So in other words, a quadratic function's end behavior – the arms – are pointing in the same direction.

Take your arms and make a big U! This function is simply $f(x)=x^{2}$. Everybody got it? OK, let's transform this function. Let's reflect this function on the x-axis. $f(x)=-x^{2}$ Take a look at Janice's graph. If you take a pen and place it on the x-axis you see that the orginal function points in the positive y direction and the transformed function points in the negative y direction.

Now let's look at $f(x)={\sqrt {x}}$. It looks like it's on its side, doesn't it? Let's apply more transformations. Reflection on the x-axis. Now reflect on the y-axis. Now make a reflection on both the x and y-axis. Everybody see what happens? Keep practising at home for the next lesson, OK guys?

Cubical cube farms!

deflection: a straight line with a bendy curve in the middle

OK now that we touched on the linear and quadratic functions, let's look at a very basic cubic function.

$f(x)=x^{3}$

Notice the end behaviors and the range now. Similar to a linear, right? But there's a difference! There is a deflection in this function! Make sure to show this on your tests and quizzes.

Be rational

rational function: polynomial over polynomial asymptote: parts of your body where you can't reach

Does any body know what a rational function is? No one? Well, a rational function is a function that's in fraction. So would ${\frac {1}{x}}$ be a rational function? Yes, of course! But what do you notice about it? That's right, there's parts of the graph where you can't reach. They are called asymptotes. Take a look at ${\frac {1}{x^{2}}}$. See the differences in their end behaviour.

Now let's graph this function! It's a bit tricky of the asymptotes, but you should be able to bend your elbows so you show a smooth curve. Remember now, you can't bend your arms to some positions. In some cases you can, such as a hole, but it's not necessary to know that.

Anyway that sums up unit 1. Time goes by quickly, doesn't it? But don't relax yet – we still have 2 units to go!

Unit 2

P