0.8 Exponential functions and graphs

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 $y=a{b}^{(x+p)}+q$ 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 $y=a{b}^{(x+p)}+q$

1. On the same set of axes, with -5\le x\le 3 and -35\le y\le 35 , plot the following graphs:
   1. $f\left(x\right)=-2·2^{(x+1)}+1$
   2. $g\left(x\right)=-1·2^{(x+1)}+1$
   3. $h\left(x\right)=0·2^{(x+1)}+1$
   4. $j\left(x\right)=1·2^{(x+1)}+1$
   5. $k\left(x\right)=2·2^{(x+1)}+1$
   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.
2. On the same set of axes, with -3\le x\le 3 and -5\le y\le 20 , plot the following graphs:
   1. $f\left(x\right)=1·2^{(x+1)}-2$
   2. $g\left(x\right)=1·2^{(x+1)}-1$
   3. $h\left(x\right)=1·2^{(x+1)}+0$
   4. $j\left(x\right)=1·2^{(x+1)}+1$
   5. $k\left(x\right)=1·2^{(x+1)}+2$
   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.
3. On the same set of axes, with -5\le x\le 3 and -35\le y\le 35 , plot the following graphs:
   1. $f\left(x\right)=-2·2^{(x+1)}+1$
   2. $g\left(x\right)=-1·2^{(x+1)}+1$
   3. $h\left(x\right)=0·2^{(x+1)}+1$
   4. $j\left(x\right)=1·2^{(x+1)}+1$
   5. $k\left(x\right)=2·2^{(x+1)}+1$
   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.
4. Following the general method of the above activities, choose your own values of $a$ and $q$ to plot 5 graphs of $y=a{b}^{(x+p)}+q$ 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] .

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

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 $a$ .

If $a>0$ then:

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 $q$ .

If $a<0$ then: