# 4.4 Exponential functions  (Page 2/2)

 Page 2 / 2

## Exponential functions and graphs

1. Draw the graphs of $y={2}^{x}$ and $y={\left(\frac{1}{2}\right)}^{x}$ 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 ${2}^{x}={\left(\frac{1}{2}\right)}^{x}$ 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

• Charecteristics of functions 1. dependent and independent variables2. domain and range 3. intercepts with axes4. turning points 5. asymptotes6. lines of symmetry 7. intervals on which the function increases/decreases8. continuous nature of the function
• Dependent and independent variablesThe 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.
• Domain and range 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.
• 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’
• Intercepts with the axes 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.
• Turning points Only for graphs of functions whose highest power is more than 1There 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.
• Asymptote An asymptote is a straight or curved line, which the graph of a function will approach, but never touch.
• Lines of symmetry A line about which the graph is symmetric
• Decreasing/increasing intervals The interval on which a graph increases or decreases
• Discrete and continuous nature A graph is said to be continuous if there are no breaks in the graph.
• Functions of the form y = ax + q Straight lineTable summarising a and q domain and range
• Functions of the Form y = ax^2 + q ParabolaTable summarising a and q Domain is fixed, range variesTurning point present Axis of symmetry through turning point
• Functions of the Form y = a/x+q Hyperbolic functionsTable summarising a and q Domain and rangeAsymptotes
• Functions of the Form y = ab^(x) + q Exponential functionsTable summarising a and q Domain fixed, range variesAsymptote

## End of chapter exercises

1. Given the functions $f\left(x\right)=-2{x}^{2}-18$ and $g\left(x\right)=-2x+6$
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. $f\left(x\right)>0$
2. $\frac{f\left(x\right)}{g\left(x\right)}\le 0$
4. Give the equation of the reflection of $f$ in the $x$ -axis.
2. After a ball is dropped, the rebound height of each bounce decreases. The equation $y=5·{\left(0,8\right)}^{x}$ 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 ?
3. 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?

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
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
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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
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 ?
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
Got questions? Join the online conversation and get instant answers!