Multiple transformations

 

A general exponential curve

The exponential curve given by $y=A\times b^{mx+c}+k$y=A×bmx+c+k represents a transformation of the basic curve $y=b^x$y=bx. Introducing constants enables the model to become a powerful tool in the investigation of certain types of growth and decay phenomena. Modelling with theoretical functions in this way provides a great example of why the study of mathematics is so crucial to our understanding of nature. 

Examples 

The functions $y=2^{5x+3}$y=25x+3 and $y=120\times2^{-x}$y=120×2−x are examples of the general exponential function given by $y=A\times b^{mx+c}+k$y=A×bmx+c+k, with both $A$A and $m$m non-zero. The number $b$b is known as the base of the function, and it is strictly defined as a positive number not equal to 1. 

For $y=2^{5x+3}$y=25x+3, we would say that $A=1$A=1, $b=2$b=2, $m=5$m=5, $c=3$c=3 and $k=0$k=0. For $y=120\times2^{-x}$y=120×2−x, we would say that $A=120$A=120, $b=2$b=2, $m=-1$m=−1, $c=0$c=0 and $k=0$k=0.

The function $y=4-2\times\left(0.5\right)^{-x}$y=4−2×(0.5)−x has $A=-2$A=−2, $b=0.5$b=0.5, $m=-1$m=−1, $c=0$c=0 and $k=4$k=4. Each constant has a particular effect on the overall graph.   

The constant $m$m is often called the growth constant (or decay constant if $m$m is negative). It can take on a range of non-zero values designed to suit particular real life growth or decay rates. However, for our immediate purposes, we will restrict $m$m to non-zero integer values only.

We introduce all of these constants in order to accurately model real world phenomena. This is the power of a generalised model. We can adjust the constants to fit reality and in so doing learn more about the way nature works.

A few key points

Whilst the general form is a comprehensive tool for sketching exponential curves, there are a few simpler observations to keep in mind. We can summarise them using examples as shown in this table:

Specific Example	Observation
$y=-3^x$y=−3x	Reflect $y=3^x$y=3x across the $x$x-axis
$y=3^{x-5}$y=3x−5	Translate $y=3^x$y=3x horizontally  to the right by $5$5 units
$y=3^x-5$y=3x−5	Translate $y=3^x$y=3x vertically downward by $5$5 units
$y=2\times3^x$y=2×3x	Double every $y$y value of  $y=3^x$y=3x
$y=8-3^x$y=8−3x	Reflect $y=3^x$y=3x across the $x$x axis then translate $8$8 units upward

More complex forms of the exponential require more thought. For example, the function $y=3^{2x-5}$y=32x−5 is quite interesting to think about. The applet below can produce the graph as a plot of points, but we can think about what the curve might look like without it.

For example, we can rewrite the function as follows:

 $y=3^{2\left(x-\frac{5}{2}\right)}=\left(3^2\right)^{\left(x-\frac{5}{2}\right)}=9^{\left(x-\frac{5}{2}\right)}.$y=32(x−52​)=(32)(x−52​)=9(x−52​).

Hence, the function could be thought of as the function $y=9^x$y=9x translated to the right by $2\frac{1}{2}$212​ units.

 

The applet

The applet below is extremely versatile, but we need to keep in mind that it is a learning tool exploring the effects of the different constants involved. As a guide, it might be helpful to use the applet to create the four graphs shown in this table.

Verify the $y$y-intercepts of each graph, the limiting value of $y$y (this is the value that the function gets close to without actually ever reaching) and whether or not the graph is rising or falling:

Function	$y$y-intercept	limiting value	rising/falling
$y=2^x$y=2x	$y=1$y=1	$y=0$y=0	rising
$y=3^{-x+1}$y=3−x+1	$y=1$y=1	$y=0$y=0	falling
$y=3\times4^x-2$y=3×4x−2	$y=1$y=1	$y=-2$y=−2	rising
$y=\left(0.5\right)^x$y=(0.5)x	$y=1$y=1	$y=0$y=0	falling

After experimenting with these, try other combinations of constants. What can you learn? 

 

BASES BETWEEN 0 AND 1

One final point that should be noted is that a curve like $y=\left(0.5\right)^x$y=(0.5)x is none other than $y=2^{-x}$y=2−x in disguise. Thus: 

$y=\left(0.5\right)^x=\left(\frac{1}{2}\right)^x=\frac{1}{2^x}=2^{-x}$y=(0.5)x=(12​)x=12x​=2−x 

 

In a similar way we can say that $y=\left(\frac{1}{b}\right)^x=b^{-x}$y=(1b​)x=b−x, and so every exponential curve of the form $y=b^x$y=bx, with a base $b$b in the interval $00<b<1, can be re-expressed as $y=\left(\frac{1}{b}\right)^{-x}$y=(1b​)−x. Since $b$b is a positive number, this means that exponential functions of the form  $y=b^x$y=bx where $00<b<1 are in fact decreasing curves.  

Worked Examples

Question 1

Question 2

Question 3