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
$q$ .
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