<< Chapter < Page Chapter >> Page >
g ( x ) = ( x - 1 ) 2 + 2 y i n t = ( 0 - 1 ) 2 + 2 = ( - 1 ) 2 + 2 = 1 + 2 = 3

The x -intercepts are calculated as follows:

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

However, [link] is only valid if - q a > 0 which means that either q < 0 or a < 0 but not both. 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 with turning point above the x -axis 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 twice.

For example, the x -intercepts of g ( x ) = ( x - 1 ) 2 + 2 are given by setting y = 0 to get:

g ( x ) = ( x - 1 ) 2 + 2 0 = ( x i n t - 1 ) 2 + 2 - 2 = ( x i n t - 1 ) 2

which has no real solutions. Therefore, the graph of g ( x ) = ( x - 1 ) 2 + 2 does not have any x -intercepts.

Intercepts

  1. Find the x- and y-intercepts of the function f ( x ) = ( x - 4 ) 2 - 1 .
  2. Find the intercepts with both axes of the graph of f ( x ) = x 2 - 6 x + 8 .
  3. Given: f ( x ) = - x 2 + 4 x - 3 . Calculate the x- and y-intercepts of the graph of f .

Turning points

The turning point of the function of the form f ( x ) = a ( x + p ) 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 + p ) 2 + q is { f ( x ) : f ( x ) [ q , ) } and if a < 0 then the range of f ( x ) = a ( x + p ) 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 + p ) 2 + q 0 = a ( x + p ) 2 0 = ( x + p ) 2 0 = x + p x = - p

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

Turning points

  1. Determine the turning point of the graph of f ( x ) = x 2 - 6 x + 8 .
  2. Given: f ( x ) = - x 2 + 4 x - 3 . Calculate the co-ordinates of the turning point of f .
  3. Find the turning point of the following function by completing the square: y = 1 2 ( x + 2 ) 2 - 1 .

Axes of symmetry

There is only one axis of symmetry for the function of the form f ( x ) = a ( x + p ) 2 + q . This axis passes through the turning point and is parallel to the y -axis. Since the x -coordinate of the turning point is x = - p , this is the axis of symmetry.

Axes of symmetry

  1. Find the equation of the axis of symmetry of the graph y = 2 x 2 - 5 x - 18 .
  2. Write down the equation of the axis of symmetry of the graph of y = 3 ( x - 2 ) 2 + 1 .
  3. Write down an example of an equation of a parabola where the y-axis is the axis of symmetry.

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

In order to sketch graphs of the form f ( x ) = a ( x + p ) 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 + 1 ) 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 + 1 ) 2 0 - 1 2 ( x + 1 ) 2 0 - 1 2 ( x + 1 ) 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 + 1 ) 2 - 3 = - 3 - 1 2 ( x + 1 ) 2 - 3 + 3 = 0 - 1 2 ( x + 1 ) 2 = 0 Divide both sides by - 1 2 : ( x + 1 ) 2 = 0 Take square root of both sides: x + 1 = 0 x = - 1

The coordinates of the turning point are: ( - 1 , - 3 ) .

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

y i n t = - 1 2 ( 0 + 1 ) 2 - 3 = - 1 2 ( 1 ) - 3 = - 1 2 - 3 = - 1 2 - 3 = - 7 2

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

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

which has no real solutions. Therefore, there are no x -intercepts.

We also know that the axis of symmetry is parallel to the y -axis and passes through the turning point.

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

Khan academy video on graphing quadratics

Sketching the parabola

  1. Draw the graph of y = 3 ( x - 2 ) 2 + 1 showing all the intercepts with the axes as well as the coordinates of the turning point.
  2. Draw a neat sketch graph of the function defined by y = a x 2 + b x + c if a > 0 ; b < 0 ; b 2 = 4 a c .

Writing an equation of a shifted parabola

Given a parabola with equation y = x 2 - 2 x - 3 . The graph of the parabola is shifted one unit to the right. Or else the y-axis shifts one unit to the left i.e. x becomes x - 1 . Therefore the new equation will become:

y = ( x - 1 ) 2 - 2 ( x - 1 ) - 3 = x 2 - 2 x + 1 - 2 x + 2 - 3 = x 2 - 4 x

If the given parabola is shifted 3 units down i.e. y becomes y + 3 . The new equation will be:

(Notice the x-axis then moves 3 units upwards)

y + 3 = x 2 - 2 x - 3 y = x 2 - 2 x - 6

End of chapter exercises

  1. Show that if a < 0 , then the range of f ( x ) = a ( x + p ) 2 + q is { f ( x ) : f ( x ) ( - , q ] } .
  2. If (2,7) is the turning point of f ( x ) = - 2 x 2 - 4 a x + k , find the values of the constants a and k .
  3. The graph in the figure is represented by the equation f ( x ) = a x 2 + b x . The coordinates of the turning point are (3,9). Show that a = - 1 and b = 6 .
  4. Given: y = x 2 - 2 x + 3 . Give the equation of the new graph originating if:
    1. The graph of f is moved three units to the left.
    2. The x -axis is moved down three units.
  5. A parabola with turning point (-1,-4) is shifted vertically by 4 units upwards. What are the coordinates of the turning point of the shifted parabola?

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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
Other chapter Q/A we can ask
Moahammedashifali Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Siyavula textbooks: grade 11 maths' conversation and receive update notifications?

Ask