# 0.8 Exponential functions and graphs

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## 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 $y=a{b}^{\left(x+p\right)}+q$ 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 $y=a{b}^{\left(x+p\right)}+q$

1. On the same set of axes, with $-5\le x\le 3$ and $-35\le y\le 35$ , plot the following graphs:
1. $f\left(x\right)=-2·{2}^{\left(x+1\right)}+1$
2. $g\left(x\right)=-1·{2}^{\left(x+1\right)}+1$
3. $h\left(x\right)=0·{2}^{\left(x+1\right)}+1$
4. $j\left(x\right)=1·{2}^{\left(x+1\right)}+1$
5. $k\left(x\right)=2·{2}^{\left(x+1\right)}+1$
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. $f\left(x\right)=1·{2}^{\left(x+1\right)}-2$
2. $g\left(x\right)=1·{2}^{\left(x+1\right)}-1$
3. $h\left(x\right)=1·{2}^{\left(x+1\right)}+0$
4. $j\left(x\right)=1·{2}^{\left(x+1\right)}+1$
5. $k\left(x\right)=1·{2}^{\left(x+1\right)}+2$
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. $f\left(x\right)=-2·{2}^{\left(x+1\right)}+1$
2. $g\left(x\right)=-1·{2}^{\left(x+1\right)}+1$
3. $h\left(x\right)=0·{2}^{\left(x+1\right)}+1$
4. $j\left(x\right)=1·{2}^{\left(x+1\right)}+1$
5. $k\left(x\right)=2·{2}^{\left(x+1\right)}+1$
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 $y=a{b}^{\left(x+p\right)}+q$ 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 $y=a{b}^{\left(x+p\right)}+q$ , the function is defined for all real values of $x$ . Therefore, the domain is $\left\{x:x\in \mathbb{R}\right\}$ .

The range of $y=a{b}^{\left(x+p\right)}+q$ is dependent on the sign of $a$ .

If $a>0$ then:

$\begin{array}{ccc}\hfill {b}^{\left(x+p\right)}& >& 0\hfill \\ \hfill a·{b}^{\left(x+p\right)}& >& 0\hfill \\ \hfill a·{b}^{\left(x+p\right)}+q& >& q\hfill \\ \hfill f\left(x\right)& >& q\hfill \end{array}$

Therefore, if $a>0$ , then the range is $\left\{f\left(x\right):f\left(x\right)\in \left[q,\infty \right)\right\}$ . In other words $f\left(x\right)$ can be any real number greater than $q$ .

If $a<0$ then:

$\begin{array}{ccc}\hfill {b}^{\left(x+p\right)}& >& 0\hfill \\ \hfill a·{b}^{\left(x+p\right)}& <& 0\hfill \\ \hfill a·{b}^{\left(x+p\right)}+q& <& q\hfill \\ \hfill f\left(x\right)& <& q\hfill \end{array}$

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