# 0.6 Quadratic functions and graphs

 Page 1 / 2

## Introduction

In Grade 10, you studied graphs of many different forms. In this chapter, you will learn a little more about the graphs of quadratic functions.

## Functions of the form $y=a{\left(x+p\right)}^{2}+q$

This form of the quadratic function is slightly more complex than the form studied in Grade 10, $y=a{x}^{2}+q$ . The general shape and position of the graph of the function of the form $f\left(x\right)=a{\left(x+p\right)}^{2}+q$ is shown in [link] .

## Investigation : functions of the form $y=a{\left(x+p\right)}^{2}+q$

1. On the same set of axes, plot the following graphs:
1. $a\left(x\right)={\left(x-2\right)}^{2}$
2. $b\left(x\right)={\left(x-1\right)}^{2}$
3. $c\left(x\right)={x}^{2}$
4. $d\left(x\right)={\left(x+1\right)}^{2}$
5. $e\left(x\right)={\left(x+2\right)}^{2}$
Use your results to deduce the effect of $p$ .
2. On the same set of axes, plot the following graphs:
1. $f\left(x\right)={\left(x-2\right)}^{2}+1$
2. $g\left(x\right)={\left(x-1\right)}^{2}+1$
3. $h\left(x\right)={x}^{2}+1$
4. $j\left(x\right)={\left(x+1\right)}^{2}+1$
5. $k\left(x\right)={\left(x+2\right)}^{2}+1$
Use your results to deduce the effect of $q$ .
3. Following the general method of the above activities, choose your own values of $p$ and $q$ to plot 5 different graphs (on the same set of axes) of $y=a{\left(x+p\right)}^{2}+q$ to deduce the effect of $a$ .

From your graphs, you should have found that $a$ affects whether the graph makes a smile or a frown. If $a<0$ , the graph makes a frown and if $a>0$ then the graph makes a smile. This was shown in Grade 10.

You should have also found that the value of $q$ affects whether the turning point of the graph is above the $x$ -axis ( $q<0$ ) or below the $x$ -axis ( $q>0$ ).

You should have also found that the value of $p$ affects whether the turning point is to the left of the $y$ -axis ( $p>0$ ) or to the right of the $y$ -axis ( $p<0$ ).

These different properties are summarised in [link] . The axes of symmetry for each graph is shown as a dashed line.

 $p<0$ $p>0$ $a>0$ $a<0$ $a>0$ $a<0$ $q\ge 0$ $q\le 0$

## Domain and range

For $f\left(x\right)=a{\left(x+p\right)}^{2}+q$ , the domain is $\left\{x:x\in \mathbb{R}\right\}$ because there is no value of $x\in \mathbb{R}$ for which $f\left(x\right)$ is undefined.

The range of $f\left(x\right)=a{\left(x+p\right)}^{2}+q$ depends on whether the value for $a$ is positive or negative. We will consider these two cases separately.

If $a>0$ then we have:

$\begin{array}{ccc}\hfill {\left(x+p\right)}^{2}& \ge & 0\phantom{\rule{1.em}{0ex}}\left(\mathrm{The square of an expression is always positive}\right)\hfill \\ \hfill a{\left(x+p\right)}^{2}& \ge & 0\phantom{\rule{1.em}{0ex}}\left(\mathrm{Multiplication by a positive number maintains the nature of the inequality}\right)\hfill \\ \hfill a{\left(x+p\right)}^{2}+q& \ge & q\hfill \\ \hfill f\left(x\right)& \ge & q\hfill \end{array}$

This tells us that for all values of $x$ , $f\left(x\right)$ is always greater than or equal to $q$ . Therefore if $a>0$ , the range of $f\left(x\right)=a{\left(x+p\right)}^{2}+q$ is $\left\{f\left(x\right):f\left(x\right)\in \left[q,\infty \right)\right\}$ .

Similarly, it can be shown that if $a<0$ that the range of $f\left(x\right)=a{\left(x+p\right)}^{2}+q$ is $\left\{f\left(x\right):f\left(x\right)\in \left(-\infty ,q\right]\right\}$ . This is left as an exercise.

For example, the domain of $g\left(x\right)={\left(x-1\right)}^{2}+2$ is $\left\{x:x\in \mathbb{R}\right\}$ because there is no value of $x\in \mathbb{R}$ for which $g\left(x\right)$ is undefined. The range of $g\left(x\right)$ can be calculated as follows:

$\begin{array}{ccc}\hfill {\left(x-p\right)}^{2}& \ge & 0\hfill \\ \hfill {\left(x+p\right)}^{2}+2& \ge & 2\hfill \\ \hfill g\left(x\right)& \ge & 2\hfill \end{array}$

Therefore the range is $\left\{g\left(x\right):g\left(x\right)\in \left[2,\infty \right)\right\}$ .

## Domain and range

1. Given the function $f\left(x\right)={\left(x-4\right)}^{2}-1$ . Give the range of $f\left(x\right)$ .
2. What is the domain of the equation $y=2{x}^{2}-5x-18$ ?

## Intercepts

For functions of the form, $y=a{\left(x+p\right)}^{2}+q$ , the details of calculating the intercepts with the $x$ and $y$ axes is given.

The $y$ -intercept is calculated as follows:

$\begin{array}{ccc}\hfill y& =& a{\left(x+p\right)}^{2}+q\hfill \\ \hfill {y}_{int}& =& a{\left(0+p\right)}^{2}+q\hfill \\ & =& a{p}^{2}+q\hfill \end{array}$

If $p=0$ , then ${y}_{int}=q$ .

For example, the $y$ -intercept of $g\left(x\right)={\left(x-1\right)}^{2}+2$ is given by setting $x=0$ to get:

what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Other chapter Q/A we can ask