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Graphs of exponential functions

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Mathematics Teaching for Learning
Engineering Maths B, Exponents, Number
August 18th, 2008 by Craig Rose
Permalink: http://www.craigsmaths.com/number/graphs-of-exponential-functions/

Engineering Maths B, Exponents, Number
August 18th, 2008 by Craig Rose
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By comparing exponential and logarithmic functions with their simplest forms it is easy to sketch their graphs using a transformational approach.

All graphs in this article were created with OpenPlaG.

The simple exponential functions

First take a look at the simplest exponential function and it’s graph:

$y=e^{x}$

Simple exponential graph

This function and graph has the following properties for all values of $e>1$ :

All values of $y$ are positive
The y-intercept (where $x=0$ ) is at $y=1$
The graph has an asymptote at $y=0$
As $x$ increases $y$ increases rapidly (exponentially) to $\infty$
The rate of increase in $y$ is dependant on $e$

For purposes of comparison we can take $e$ to be equal to Euler’s number. This is useful for many real world applications and is unique in that the derivative of $e^{x}$ is equal to itself for all $x$ . ie $\frac{d}{dx}e^{x}=e^{x}$ .

The general exponential functions

The general form can be expressed as:

$y=ae^{b\left(x-c\right)}+d$

Comparing this with the simple case we can observe the various transformations resulting:

Dilation in the y direction ( $a$ )

$a$ is a dilation (stretched or compressed) in the y-direction. For $0<\left|a\right|<1$ the graph will be compressed and for $\left|a\right|>1$ it will stretch. If $a<0$ then the graph will dilate and be reflected in the x-axis.

Examples:

Exponential function with positive y-dilation

Exponential function with negative y dilation

Dilation in the x direction ( $b$ )

$b$ is a dilation (stretched or compressed) in the x-direction. For $0<\left|b\right|<1$ the graph will be compressed and for $\left|a\right|>1$ it will stretch. If $b<0$ then the graph will dilate and be reflected in the y-axis.

Exponential function with positive x-dilation

Exponential function with negative y-dilation

Translation in the x-direction ( $c$ )

When $c>0$ the entire graph will translate $\left|c\right|$ units to the right. When $c<0$ the entire graph will translate $\left|c\right|$ units to the left.

Translation in the x direction

Translation in the y-direction ( $d$ )

When $d>0$ the entire graph will translate $\left|d\right|$ units up. When $d<0$ the entire graph will translate $\left|d\right|$ units down.

Translation in the y direction

Sketching the graphs

Get the function into the general form

You want to end up with a coefficient of $x$ equal to $1$ .

Find the y intercept

There will always be a y-intercept. This will be where $x=0$ . Simply calculate the y-intercept using a calculator if the arithmetic requires it.

Is there an x-intercept?

An x-intercept will exist if putting $y=0$ results in a function with a real solution.

$0=ae^{b\left(x-c\right)}+d$

Solving for $x$ results in:

$x=\frac{\ln\left(-d\right)}{b\left(ln\left(a\right)+1\right)}+c$

Since $\ln\left(-d\right)$ only has solutions for $d<0$ we can tell immediately from the general form that there will only be an x-intercept if $d<0$ .

Find the x intercept.

There is no need to memorise the equation derived in the previous step. Take your original function, put $y=0$ then solve for $x$ . In the process you will want to take a logarithm of both sides of the equation.

Find the asymptote.

The asymptote occurs when $\mathop {\lim }\limits_{\left|x\right| \to \infty } ae^{b\left(x-c\right)} = 0$

This results in the asymptote being:

$y=d$

Apply any reflections

Recall that $a<0$ will reflect the simple exponential in the x-axis and $b<0$ is the reflection in the y-axis.

Sketch the graph

Now you have enough information to do the sketch. To give a better concept of the shape of the graph you may want to sketch in some calculated points as well. One at $y=e$ (the base) would be useful.