NZ Level 7 (NZC) Level 2 (NCEA)
Transformations of Exponential graphs
Lesson

## 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=120×2x 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=120×2x, 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=42×(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=3x5 Translate $y=3^x$y=3x horizontally  to the right by $5$5 units
$y=3^x-5$y=3x5 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=83x 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=32x5 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(x52)=(32)(x52)=9(x52).

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=3x+1 $y=1$y=1 $y=0$y=0 falling
$y=3\times4^x-2$y=3×4x2 $y=1$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=2x 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=2x

In a similar way we can say that $y=\left(\frac{1}{b}\right)^x=b^{-x}$y=(1b)x=bx, 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

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

Identify the:

1. Horizontal translation

2. Vertical translation

3. Growth constant

##### Question 2

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

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

2. Find the equation of the inverse function of $y=10^{-x}$y=10x.

3. Hence determine the correct graph for $y=-\log x$y=logx.

A

B

C

D

A

B

C

D

##### Question 3

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

1. Find the $y$y-intercept of the curve $y=3^{-x}$y=3x.

2. Fill in the table of values for $y=3^{-x}$y=3x.

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

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

5. Is the function $y=3^{-x}$y=3x, an increasing or decreasing function?

Increasing function

A

Decreasing function

B

Increasing function

A

Decreasing function

B

### Outcomes

#### M7-2

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

#### 91257

Apply graphical methods in solving problems