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
$q$ .
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
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?