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
$q$ .
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
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