St. Paul Math TalkConsultation on Functions

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

How about the next one?

Linear function

𝑦=3 𝑥+57 Is also a linear function because

you can write it as

Linear Function

Now let us generate a few of the pairs of numbers described by this

function.

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

Aren’t you curious why functions in the form

Y=ax+bAre called linear functions?

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

Figure 1.1 Graph y = 4x + 5

Graph of linear functions

Figure 1.2 Graph y = (3x+5)/7

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

Ano yun?????

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 function

>>>>>a = 1, b = 0, c = 0

>>>>>a = 1, b = 0, c = 5

>>>>>a = 1, b = 5, c = 7

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.

Graphs Quadratic Functions

𝑦=−𝑥2

𝑦=−𝑥2+5 𝑥−7

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

If a>0, the graph opens upward and the range is the set of real

numbers y, such that

𝑦 ≥ 4 𝑎𝑐−𝑏2

4𝑎

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

Range Quadratic function

The letters, a, b, and c are the coefficients in the expression

Range quadratic function𝑦=𝑥2+5𝑥−7

𝑦=4 𝑎𝑐−𝑏2

4𝑎=−53

4

𝑅𝑎𝑛𝑔𝑒 : 𝑦 ≥−534

(x,y)

Range quadratic function

What is the range of

?(x,y)

𝑦=4 𝑎𝑐−𝑏2

4𝑎=3316≈1.03

Range

Assignment Functions

I will send my explanation about your assignment

tomorrow.