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
$y=a{b}^{(x+p)}+q$ 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
$y=a{b}^{(x+p)}+q$
On the same set of axes, with
-5\le x\le 3 and
-35\le y\le 35 , plot the following graphs:
$f\left(x\right)=-2\xb7{2}^{(x+1)}+1$
$g\left(x\right)=-1\xb7{2}^{(x+1)}+1$
$h\left(x\right)=0\xb7{2}^{(x+1)}+1$
$j\left(x\right)=1\xb7{2}^{(x+1)}+1$
$k\left(x\right)=2\xb7{2}^{(x+1)}+1$
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.
On the same set of axes, with
-3\le x\le 3 and
-5\le y\le 20 , plot the following graphs:
$f\left(x\right)=1\xb7{2}^{(x+1)}-2$
$g\left(x\right)=1\xb7{2}^{(x+1)}-1$
$h\left(x\right)=1\xb7{2}^{(x+1)}+0$
$j\left(x\right)=1\xb7{2}^{(x+1)}+1$
$k\left(x\right)=1\xb7{2}^{(x+1)}+2$
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.
On the same set of axes, with
-5\le x\le 3 and
-35\le y\le 35 , plot the following graphs:
$f\left(x\right)=-2\xb7{2}^{(x+1)}+1$
$g\left(x\right)=-1\xb7{2}^{(x+1)}+1$
$h\left(x\right)=0\xb7{2}^{(x+1)}+1$
$j\left(x\right)=1\xb7{2}^{(x+1)}+1$
$k\left(x\right)=2\xb7{2}^{(x+1)}+1$
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.
Following the general method of the above activities, choose your own values of
$a$ and
$q$ to plot 5 graphs of
$y=a{b}^{(x+p)}+q$ 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] .
Table summarising general shapes and positions of functions of the form
$y=a{b}^{(x+p)}+q$ .
$p<0$
$p>0$
$a>0$
$a<0$
$a>0$
$a<0$
$q>0$
$q<0$
Domain and range
For
$y=a{b}^{(x+p)}+q$ , the function is defined for all real values of
$x$ . Therefore, the domain is
$\{x:x\in \mathbb{R}\}$ .
The range of
$y=a{b}^{(x+p)}+q$ is dependent on the sign of
$a$ .
Therefore, if
$a>0$ , then the range is
$\left\{f\right(x):f(x)\in [q,\infty \left)\right\}$ . In other words
$f\left(x\right)$ can be any real number greater than
$q$ .
Economics is important because it helps people understand how a variety of factors work with and against each other to control how resources such as labor and capital get used, and how inflation, supply, demand, interest rates and other factors determine how much you pay for goods and services.
Muhammad
explain the steps taken by the government in developing rural market?
can you tell how can i economics honurs(BSC) in reputed college?
Soumya
through hard study and performing well than expected from you
Mitiku
what should i prepare for it?
Soumya
prepare first, in psychologically as well as potentially to sacrifice what's expected from you, when I say this I mean that you have to be ready, for every thing and to accept failure as a good and you need to change them to potential for achievement of ur goals
Mitiku
parna kya hai behencho?
Soumya
Hallo
Rabindranath
Hello, dear what's up?
Mitiku
cool
Momoh
good morning
Isaac
pls, is anyone here from Ghana?
Isaac
Hw s every one please
Afran
Ys please I'm in Ghana
Afran
Hello
OLANIYI
pls anyone from Nigeria
OLANIYI
am a new candidate here, can someone put me 2ru
OLANIYI
hello
OLANIYI
Pls economic A level exam tomorrow pls help me
akinwale
am from Ghana
Jacob
Pls economic A level exam tomorrow pls help me
akinwale
Hi
Dev
bol Diya discuss
ab krega v
Dev
hello Mr. Rabindranath
Dev
what do you want Dimlare
Dev
yes tell me your desire to have it
Dev
to have what?
OLANIYI
Good luck
JOSEPH
I want to know about economic A level tomorrow pls help
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.