Exponentials

Hong Kong

Stage 4 - Stage 5

Lesson

The exponential curve given by $y=A\times b^{mx+c}+k$`y`=`A`×`b``m``x`+`c`+`k` represents a transformation of the basic curve $y=b^x$`y`=`b``x`. 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.

The functions $y=2^{5x+3}$`y`=25`x`+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`×`b``m``x`+`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`=25`x`+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.

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`=32`x`−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`=9`x` translated to the right by $2\frac{1}{2}$212 units.

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?

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`=12`x`=2−`x`

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

The function $y=5^x$`y`=5`x` has been transformed into the function $y=5^{x+4}+2$`y`=5`x`+4+2

Identify the:

Horizontal translation

Vertical translation

Growth constant

Consider the function $y=10^{-x}$`y`=10−`x` and its inverse function:

Plot the graph of $y=10^{-x}$

`y`=10−`x`.Loading Graph...Find the equation of the inverse function of $y=10^{-x}$

`y`=10−`x`.Hence determine the correct graph for $y=-\log x$

`y`=−`l``o``g``x`.Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...D

Consider the function $y=3^{-x}$`y`=3−`x` :

Find the $y$

`y`-intercept of the curve $y=3^{-x}$`y`=3−`x`.Fill in the table of values for $y=3^{-x}$

`y`=3−`x`.$x$ `x`$-3$−3 $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $3$3 $y$ `y`$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Find the horizontal asymptote of the curve $y=3^{-x}$

`y`=3−`x`.Hence plot the curve $y=3^{-x}$

`y`=3−`x`.Loading Graph...Is the function $y=3^{-x}$

`y`=3−`x`, an increasing or decreasing function?Increasing function

ADecreasing function

B