# 0.8 Exponential functions and graphs

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## Introduction

In Grade 10, you studied graphs of many different forms. In this chapter, you will learn a little more about the graphs of exponential functions.

## Functions of the form $y=a{b}^{\left(x+p\right)}+q$ For $b>0$

This form of the exponential function is slightly more complex than the form studied in Grade 10.

## Investigation : functions of the form $y=a{b}^{\left(x+p\right)}+q$

1. On the same set of axes, with $-5\le x\le 3$ and $-35\le y\le 35$ , plot the following graphs:
1. $f\left(x\right)=-2·{2}^{\left(x+1\right)}+1$
2. $g\left(x\right)=-1·{2}^{\left(x+1\right)}+1$
3. $h\left(x\right)=0·{2}^{\left(x+1\right)}+1$
4. $j\left(x\right)=1·{2}^{\left(x+1\right)}+1$
5. $k\left(x\right)=2·{2}^{\left(x+1\right)}+1$
Use your results to understand what happens when you change the value of $a$ . You should find that the value of $a$ affects whether the graph curves upwards ( $a>0$ ) or curves downwards ( $a<0$ ). You should also find that a larger value of $a$ (when $a$ is positive) stretches the graph upwards. However, when $a$ is negative, a lower value of $a$ (such as -2 instead of -1) stretches the graph downwards. Finally, note that when $a=0$ the graph is simply a horizontal line. This is why we set $a\ne 0$ in the original definition of these functions.
2. On the same set of axes, with $-3\le x\le 3$ and $-5\le y\le 20$ , plot the following graphs:
1. $f\left(x\right)=1·{2}^{\left(x+1\right)}-2$
2. $g\left(x\right)=1·{2}^{\left(x+1\right)}-1$
3. $h\left(x\right)=1·{2}^{\left(x+1\right)}+0$
4. $j\left(x\right)=1·{2}^{\left(x+1\right)}+1$
5. $k\left(x\right)=1·{2}^{\left(x+1\right)}+2$
Use your results to understand what happens when you change the value of $q$ . You should find that when $q$ is increased, the whole graph is translated (moved) upwards. When $q$ is decreased (poosibly even made negative), the graph is translated downwards.
3. On the same set of axes, with $-5\le x\le 3$ and $-35\le y\le 35$ , plot the following graphs:
1. $f\left(x\right)=-2·{2}^{\left(x+1\right)}+1$
2. $g\left(x\right)=-1·{2}^{\left(x+1\right)}+1$
3. $h\left(x\right)=0·{2}^{\left(x+1\right)}+1$
4. $j\left(x\right)=1·{2}^{\left(x+1\right)}+1$
5. $k\left(x\right)=2·{2}^{\left(x+1\right)}+1$
Use your results to understand what happens when you change the value of $a$ . You should find that the value of $a$ affects whether the graph curves upwards ( $a>0$ ) or curves downwards ( $a<0$ ). You should also find that a larger value of $a$ (when $a$ is positive) stretches the graph upwards. However, when $a$ is negative, a lower value of $a$ (such as -2 instead of -1) stretches the graph downwards. Finally, note that when $a=0$ the graph is simply a horizontal line. This is why we set $a\ne 0$ in the original definition of these functions.
4. Following the general method of the above activities, choose your own values of $a$ and $q$ to plot 5 graphs of $y=a{b}^{\left(x+p\right)}+q$ on the same set of axes (choose your own limits for $x$ and $y$ carefully). Make sure that you use the same values of $a$ , $b$ and $q$ for each graph, and different values of $p$ . Use your results to understand the effect of changing the value of $p$ .

These different properties are summarised in [link] .

 $p<0$ $p>0$ $a>0$ $a<0$ $a>0$ $a<0$ $q>0$ $q<0$

## Domain and range

For $y=a{b}^{\left(x+p\right)}+q$ , the function is defined for all real values of $x$ . Therefore, the domain is $\left\{x:x\in \mathbb{R}\right\}$ .

The range of $y=a{b}^{\left(x+p\right)}+q$ is dependent on the sign of $a$ .

If $a>0$ then:

$\begin{array}{ccc}\hfill {b}^{\left(x+p\right)}& >& 0\hfill \\ \hfill a·{b}^{\left(x+p\right)}& >& 0\hfill \\ \hfill a·{b}^{\left(x+p\right)}+q& >& q\hfill \\ \hfill f\left(x\right)& >& q\hfill \end{array}$

Therefore, if $a>0$ , then the range is $\left\{f\left(x\right):f\left(x\right)\in \left[q,\infty \right)\right\}$ . In other words $f\left(x\right)$ can be any real number greater than $q$ .

If $a<0$ then:

$\begin{array}{ccc}\hfill {b}^{\left(x+p\right)}& >& 0\hfill \\ \hfill a·{b}^{\left(x+p\right)}& <& 0\hfill \\ \hfill a·{b}^{\left(x+p\right)}+q& <& q\hfill \\ \hfill f\left(x\right)& <& q\hfill \end{array}$

any tip for igcse economics exam?pls
well
The
What is a market
what are the variables that affect demand
Divine
what are the variables that affect demand
Divine
what are the variables that affect demand
Divine
what are the variables that affect demand
Divine
what are the variables that affect demand
Divine
price of the related goods 2 price of the given commodity 3 income of the consumer 4 taste and preference 5 expectation in the future price
John
pls the taste and preference
Nas
explain briefly
Nas
a consumer taste and preference commodity changes for a time the man becomes
John
sorry sorry
John
is when the price of a commodity becomes high and can't afford example Samsung instead of iPhone
John
consumers who have high intense for goods will purchase the goods even if the price of that commodity increases because he or she preferred that commodity.people will be prefer iphone as its price increase
Yussif
as usual bad taste of preference is when a consumer regrets from one commodity to another in terms of the price
John
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#Preference; #Income #Test
Dereje
#price Of Commodity #Income #Taste #Preference
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#Market is The Place Where Buyers And Sellers Are Exchanging Their Goods And service. #
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difference between macro and micro economics
Lawrence
Microeconomic Study about individual consumers market But Macroeconomis Study General economic Process Such As #Aggregate Demand #Aggregate Supples #GDp= #GNp
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Lawrence
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Chinex
pls can someone differentiate between the perfectly elastic, perfectly inelastic and unitary
and then again pls what are the types of elasticity, the methods of calculating it thank u
yhar
Perfectly inelastic is when the coefficient is equal to zero Unitary is when the coefficient is equal to one But am not sure if we have perfectly inelastic
John
I'm kind off confuse abt the PED, IED and co are they the types of elasticity we've
yhar
Yh the types are price elasticity cross and income elasticity of demand
John
do we've specific formulaes to calculate for each of them
yhar
yes. PED. changes in quantity demanded divided by changes in price
Vealmurugan
so pls what's the general name given to unitary, elastic n inelastic ? are the names given to the final result after doing the calculations?
yhar
P2-P1÷P1×100or Q2-Q1×Q1×100 PED
John
***tutor2u.net/economics/reference/price-elasticity-of-demand
Vealmurugan
They are elasticity coefficient
John
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John
P2-P1Ã·P1Ã100or Q2-Q1ÃQ1Ã100 PED @john pls tis is what m talking abt
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John
Pls are you having a for PED
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Tayyeb
economics is a science which studies human behaviour as a relationship between ends and scarce means which have alternative use
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is a science which study human behavior as a relationship between ends and scarce means which have alternative uses
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yes this is because economic provide a body of knowledge on human economic principles under theories and these theories can be verified with real world data using science method in other words it was scientific method in arriving at solution identification of problem or basic data collection among
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unitary ElasticWhen Elasticty =1 Perfectily Elastic When 0<1 inelastic when 0>
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What is PPF
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Production Possibility Frontier
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It refers to a curve or graph which shows the possible contributions of maximum alternative of commodity that can be produced in an economy
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guys Any One With Novdec Questions 2019?
what is supply
supply relationship shows that the higher the price, the higher the quantity supplied.
what are the dissadvantages of large scale production
depreciate of quality taste
Kosiso
Some of the Disadvantages are:- 1. Production not according to individual Tastes 2. Monopoly 3. Not Flexible 4. Over-Production 5. Heavy loss and Dislocation 6. Decline of Cottage and Small Scale Industries 7. Adverse Effect on Labourers 8. Unequal Distribution of Wealth
Asrar
And what can be the advantages too
Enow
Yussif
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Enow
the country will have enough products.this will reduce the level of government expenditure on imported goods especially
Yussif
Some of Advantage 1. Division of Labour 2. More Production 3. Use of machines 4. Low Cost of Production 5. Standard Goods 6. Advertisements and Salesmanship
Asrar
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Kalu
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depression
DG
explain the following 1.supply 2.mobility of labour 3.why the demand slope downward from left to right
Mobility of labor is a the movement of labor (people) geographically or occupationally
Olotu
What is Supply
Supply is quantity of a comodity which is presented in the market for sell on fixed price
Mahwish
Supply is quality of a goods the seller are willing to deliver at the particular price
Chinex
what is economy
Economy is defined as the efficient use of resources
Thomas
explain merits and demerit of economic
Portia
What is Unemployment?
unemployment is a situation where an individual is easily and ready to render services but resources are not available.
Phelix
Unemployment refers to individual who are employable and seeking a job but are unable to find a job or doesn't have a job. To find the unemployment rate in a given economy, you must divide the unemployed people by the total number of employed people in the work force. Indicator of economy's status.
elkanah
unemployment is The level of joblessness in an economy, often measured as a percentage of the workforce. Unemployment was reported at 5.2% in May, up from 4.9% in April
ANNOR
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Olotu
just trying ⚠ factors that affect unemployment - the rate of imported inflation - interest rates - levels of investment
Wallace
unemployment is a situation where country labour Force who are active capable and are really searching for or willing to work at assistant wage rates but cannot find job
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Lawrence
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Other chapter Q/A we can ask