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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

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
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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
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
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Other chapter Q/A we can ask
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Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
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