<< Chapter < Page Chapter >> Page >

This tells us that for all values of x , f ( x ) is always greater than q . Therefore if a > 0 , the range of f ( x ) = a x 2 + q is { f ( x ) : f ( x ) [ q , ) } .

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

For example, the domain of g ( x ) = x 2 + 2 is { x : x R } because there is no value of x R for which g ( x ) is undefined. The range of g ( x ) can be calculated as follows:

x 2 0 x 2 + 2 2 g ( x ) 2

Therefore the range is { g ( x ) : g ( x ) [ 2 , ) } .

Intercepts

For functions of the form, y = a x 2 + q , the details of calculating the intercepts with the x and y axis is given.

The y -intercept is calculated as follows:

y = a x 2 + q y i n t = a ( 0 ) 2 + q = q

For example, the y -intercept of g ( x ) = x 2 + 2 is given by setting x = 0 to get:

g ( x ) = x 2 + 2 y i n t = 0 2 + 2 = 2

The x -intercepts are calculated as follows:

y = a x 2 + q 0 = a x i n t 2 + q a x i n t 2 = - q x i n t = ± - q a

However, [link] is only valid if - q a 0 which means that either q 0 or a < 0 . This is consistent with what we expect, since if q > 0 and a > 0 then - q a is negative and in this case the graph lies above the x -axis and therefore does not intersect the x -axis. If however, q > 0 and a < 0 , then - q a is positive and the graph is hat shaped and should have two x -intercepts. Similarly, if q < 0 and a > 0 then - q a is also positive, and the graph should intersect with the x -axis.

If q = 0 then we have one intercept at x = 0 .

For example, the x -intercepts of g ( x ) = x 2 + 2 is given by setting y = 0 to get:

g ( x ) = x 2 + 2 0 = x i n t 2 + 2 - 2 = x i n t 2

which is not real. Therefore, the graph of g ( x ) = x 2 + 2 does not have any x -intercepts.

Turning points

The turning point of the function of the form f ( x ) = a x 2 + q is given by examining the range of the function. We know that if a > 0 then the range of f ( x ) = a x 2 + q is { f ( x ) : f ( x ) [ q , ) } and if a < 0 then the range of f ( x ) = a x 2 + q is { f ( x ) : f ( x ) ( - , q ] } .

So, if a > 0 , then the lowest value that f ( x ) can take on is q . Solving for the value of x at which f ( x ) = q gives:

q = a x t p 2 + q 0 = a x t p 2 0 = x t p 2 x t p = 0

x = 0 at f ( x ) = q . The co-ordinates of the (minimal) turning point is therefore ( 0 , q ) .

Similarly, if a < 0 , then the highest value that f ( x ) can take on is q and the co-ordinates of the (maximal) turning point is ( 0 , q ) .

Axes of symmetry

There is one axis of symmetry for the function of the form f ( x ) = a x 2 + q that passes through the turning point. Since the turning point lies on the y -axis, the axis of symmetry is the y -axis.

Sketching graphs of the form f ( x ) = a x 2 + q

In order to sketch graphs of the form, f ( x ) = a x 2 + q , we need to determine five characteristics:

  1. sign of a
  2. domain and range
  3. turning point
  4. y -intercept
  5. x -intercept

For example, sketch the graph of g ( x ) = - 1 2 x 2 - 3 . Mark the intercepts, turning point and axis of symmetry.

Firstly, we determine that a < 0 . This means that the graph will have a maximal turning point.

The domain of the graph is { x : x R } because f ( x ) is defined for all x R . The range of the graph is determined as follows:

x 2 0 - 1 2 x 2 0 - 1 2 x 2 - 3 - 3 f ( x ) - 3

Therefore the range of the graph is { f ( x ) : f ( x ) ( - , - 3 ] } .

Using the fact that the maximum value that f ( x ) achieves is -3, then the y -coordinate of the turning point is -3. The x -coordinate is determined as follows:

- 1 2 x 2 - 3 = - 3 - 1 2 x 2 - 3 + 3 = 0 - 1 2 x 2 = 0 Divide both sides by - 1 2 : x 2 = 0 Take square root of both sides: x = 0 x = 0

The coordinates of the turning point are: ( 0 ; - 3 ) .

The y -intercept is obtained by setting x = 0 . This gives:

y i n t = - 1 2 ( 0 ) 2 - 3 = - 1 2 ( 0 ) - 3 = - 3

The x -intercept is obtained by setting y = 0 . This gives:

0 = - 1 2 x i n t 2 - 3 3 = - 1 2 x i n t 2 - 3 · 2 = x i n t 2 - 6 = x i n t 2

which is not real. Therefore, there are no x -intercepts which means that the function does not cross or even touch the x -axis at any point.

We also know that the axis of symmetry is the y -axis.

Finally, we draw the graph. Note that in the diagram only the y-intercept is marked. The graph has a maximal turning point (i.e. makes a frown) as determined from the sign of a, there are no x-intercepts and the turning point is that same as the y-intercept. The domain is all real numbers and the range is { f ( x ) : f ( x ) ( - , - 3 ] } .

Graph of the function f ( x ) = - 1 2 x 2 - 3

Draw the graph of y = 3 x 2 + 5 .

  1. The sign of a is positive. The parabola will therefore have a minimal turning point.
  2. The domain is: { x : x R } and the range is: { f ( x ) : f ( x ) [ 5 , ) } .
  3. The turning point occurs at ( 0 , q ) . For this function q = 5 , so the turning point is at ( 0 , 5 )
  4. The y-intercept occurs when x = 0 . Calculating the y-intercept gives:
    y = 3 x 2 + 5 y int = 3 ( 0 ) 2 + 5 y int = 5
  5. The x-intercepts occur when y = 0 . Calculating the x-intercept gives:
    y = 3 x 2 + 5 0 = 3 x 2 + 5 x 2 = - 3 5
    which is not real, so there are no x-intercepts.
  6. Putting all this together gives us the following graph:
Got questions? Get instant answers now!

The following video shows one method of graphing parabolas. Note that in this video the term vertex is used in place of turning point. The vertex and the turning point are the same thing.

Khan academy video on graphing parabolas - 1

Parabolas

  1. Show that if a < 0 that the range of f ( x ) = a x 2 + q is { f ( x ) : f ( x ) ( - ; q ] } .
  2. Draw the graph of the function y = - x 2 + 4 showing all intercepts with the axes.
  3. Two parabolas are drawn: g : y = a x 2 + p and h : y = b x 2 + q .
    1. Find the values of a and p .
    2. Find the values of b and q .
    3. Find the values of x for which a x 2 + p b x 2 + q .
    4. For what values of x is g increasing ?

Questions & Answers

what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how to synthesize TiO2 nanoparticles by chemical methods
Zubear
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Siyavula textbooks: grade 10 maths [caps]' conversation and receive update notifications?

Ask