St. Paul Math Talk Consultation on Functions
Aug 12, 2015
Introduction
I am happy to discover your message. I would be happy to help you out sa math
subject nyo.
Introduction
I know that there are math teachers out there who
assist students through the internet, using youtube, FB,
etc.
Introduction
Right now, I cannot duplicate their outpout. It takes time to learn
technology. And right now, this is the technology I know how to use.
Power point presentation.
Introduction
There’s a lot to talk about functions, and I am not sure if
I am imagining right your capabilities to understand
what I am saying. I am already 45 years old. But you
are so young, I don’t know what you think about, and
what your capabilities are. So eto ha, mag-eexplain na
ako….
Functions
Your topic is about functions. Its about pairs of points x, y such that
Yung ka-pair ng bawat x ay isang y lang.
Functions
Now, posible kasi na sobrang dami ng pairs of numbers
(x,y) such that its no longer practical to list them one by
one, that is why you see those equations.
Those equations will generate all possible pairs
of numbers (x,y) being described by the function
Linear function
Those equations describe the relationship between the
numbers x and y. for example lets look into the linear function
Y=4x+5Instead of listing down the pairs of numbers ,which are many, we
simply write down the equation to describe those numbers.
So if our function is the linear function defined by
Y=4x+5The pairs of numbers the arise from
this function are…..
Linear Function
x 4x+5 Y=4x+5
0 4(0)+5 5
1/2 4(1/2)+5 7
1 4(1)+5 9
-100 4(-100)+5 -395
101 4(101)+5 409
The pairs of numbers we generated so far are
(0,5), (1/2, 7), (1,9), (-100, -395), (101, 409)
These are just a few of them, there are in fact INFINITELY many of them.
Linear Function
x
1
0
-1
The pairs of numbers generated by our function
are(1, 8/7), (0, 5/7), (-1, 2/7)
But these are just a few of them. There are in fact
infinitely many of such pairs.
Linear function
They are called “linear” functions because when you plot those points on your cartesian plane (xy-plane),
what you get is a straight line.
Graph of linear functions
Even though both are in the shape of a line, they are not
identical. Zoom into the picture,
and you will discover their y and x-intercepts
are different
Figure 1.1 Graph y = 4x + 5
Figure 1.2 Graph y = (3x+5)/7
Domain and Range of Linear Functions
You may have noticed that your teacher wrote the symbols D and R. D stands for domain and R for
range of the function.
Domain and Range of Functions
Domain is the set of admissible values of
x
Range is the set of resulting values of
y
Domain and Range of Functions
Under D and R, your teacher wrote R (real
numbers)
What that means is that the set of admissible values of x are all real numbers, and the set of resulting values of y is
the set of real numbers
Domain and Range of functions
If that is domain and range, what’s the big
fuss about them? Isn’t the domain and range of ALL FUNCTIONS the set of real numbers? No, that is not true.
You will discover this with our next
example
Domain and Range of functions
Lets look into the function defined by
Get your calculator and obtain the following square
roots:
Domain and Range of functions
You got the square roots for 25 and 121 but what you got for -1 and -121 is
SYNTAX ERROR
Domain and Range of functions
It goes to show that the domain of functions is not
always the set of real numbers. The same thing must be said about range.
Quadratic Functions
A quadratic function is a function in the form
Where
The quadratic function your
teacher gave you is still a quadratic
function although the values for a, b,
and c may vary.
Quadratic functions
x y (x,y)
0 0 0 (0,0)
-1 1 (-1,1)
1 1 (1,1)
Let us look into the first quadratic function
Again there are infinitely many of such pairs of
numbers. These pairs are in fact coordinates of points.
We shall plot them
on the next page.
Graph Quadratic functions
𝑦=𝑥2
Figure 2.1 Graphs Quadratic functions
𝑦=𝑥2+5
These graphs are not identical. You can enlarge the
picture to see them closely yourself.
Graphs Quadratic functions
𝑦=𝑥2+5𝑥−7
The graph of quadratic functions are in the shape of a parabola, but not all
parabola “opens” upward.
Domain and Range of Quadratic functions
Lets go back to domain and range.
What are the domain and range
of quadratic functions?
Domain of quadratic
functions is the set of real
numbers but the range is…..
Range of Quadratic functions
The resulting values of y (range)
depends on the orientation of the
graph
Range Quadratic function
If a<0, the resulting values of y (range) is the set of real numbers such
that
𝑦 ≤ 4 𝑎𝑐−𝑏2
4𝑎