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Exponential functions and graphs

  1. Draw the graphs of y = 2 x and y = ( 1 2 ) x on the same set of axes.
    1. Is the x -axis and asymptote or and axis of symmetry to both graphs ? Explain your answer.
    2. Which graph is represented by the equation y = 2 - x ? Explain your answer.
    3. Solve the equation 2 x = ( 1 2 ) x graphically and check that your answer is correct by using substitution.
    4. Predict how the graph y = 2 . 2 x will compare to y = 2 x and then draw the graph of y = 2 . 2 x on the same set of axes.
  2. The curve of the exponential function f in the accompanying diagram cuts the y-axis at the point A(0; 1) and B(2; 4) is on f .
    1. Determine the equation of the function f .
    2. Determine the equation of h , the function of which the curve is the reflection of the curve of f in the x -axis.
    3. Determine the range of h .

Summary

  • You should know the following charecteristics of functions:
    • The given or chosen x-value is known as the independent variable, because its value can be chosen freely. The calculated y-value is known as the dependent variable, because its value depends on the chosen x-value.
    • The domain of a relation is the set of all the x values for which there exists at least one y value according to that relation. The range is the set of all the y values, which can be obtained using at least one x value.
    • The intercept is the point at which a graph intersects an axis. The x-intercepts are the points at which the graph cuts the x-axis and the y-intercepts are the points at which the graph cuts the y-axis.
    • Only for graphs of functions whose highest power is more than 1. There are two types of turning points: a minimal turning point and a maximal turning point. A minimal turning point is a point on the graph where the graph stops decreasing in value and starts increasing in value and a maximal turning point is a point on the graph where the graph stops increasing in value and starts decreasing.
    • An asymptote is a straight or curved line, which the graph of a function will approach, but never touch.
    • A line about which the graph is symmetric
    • The interval on which a graph increases or decreases
    • A graph is said to be continuous if there are no breaks in the graph.
  • Set notation A set of certain x values has the following form: {x : conditions, more conditions}
  • Interval notation Here we write an interval in the form ’lower bracket, lower number, comma, upper number, upper bracket’
  • You should know the following functions and their properties:
    • Functions of the form y = a x + q . These are straight lines.
    • Functions of the Form y = a x 2 + q These are known as parabolic functions or parabolas.
    • Functions of the Form y = a x + q . These are known as hyperbolic functions.
    • Functions of the Form y = a b ( x ) + q . These are known as exponential functions.

End of chapter exercises

  1. Sketch the following straight lines:
    1. y = 2 x + 4
    2. y - x = 0
    3. y = - 1 2 x + 2
  2. Sketch the following functions:
    1. y = x 2 + 3
    2. y = 1 2 x 2 + 4
    3. y = 2 x 2 - 4
  3. Sketch the following functions and identify the asymptotes:
    1. y = 3 x + 2
    2. y = - 4 . 2 x + 1
    3. y = 2 . 3 x - 2
  4. Sketch the following functions and identify the asymptotes:
    1. y = 3 x + 4
    2. y = 1 x
    3. y = 2 x - 2
  5. Determine whether the following statements are true or false. If the statement is false, give reasons why:
    1. The given or chosen y-value is known as the independent variable.
    2. An intercept is the point at which a graph intersects itself.
    3. There are two types of turning points – minimal and maximal.
    4. A graph is said to be congruent if there are no breaks in the graph.
    5. Functions of the form y = a x + q are straight lines.
    6. Functions of the form y = a x + q are exponential functions.
    7. An asymptote is a straight or curved line which a graph will intersect once.
    8. Given a function of the form y = a x + q , to find the y-intersect put x = 0 and solve for y .
    9. The graph of a straight line always has a turning point.
  6. Given the functions f ( x ) = - 2 x 2 - 18 and g ( x ) = - 2 x + 6
    1. Draw f and g on the same set of axes.
    2. Calculate the points of intersection of f and g .
    3. Hence use your graphs and the points of intersection to solve for x when:
      1. f ( x ) > 0
      2. f ( x ) g ( x ) 0
    4. Give the equation of the reflection of f in the x -axis.
  7. After a ball is dropped, the rebound height of each bounce decreases. The equation y = 5 · ( 0 , 8 ) x shows the relationship between x , the number of bounces, and y , the height of the bounce, for a certain ball. What is the approximate height of the fifth bounce of this ball to the nearest tenth of a unit ?
  8. Mark had 15 coins in five Rand and two Rand pieces. He had 3 more R2-coins than R5-coins. He wrote a system of equations to represent this situation, letting x represent the number of five rand coins and y represent the number of two rand coins. Then he solved the system by graphing.
    1. Write down the system of equations.
    2. Draw their graphs on the same set of axes.
    3. What is the solution?

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
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
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
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
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
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