Transformations of Exponential Graphs

 

A general exponential curve

The exponential curve given by $$y=A×bmx+c+k represents a transformation of the basic curve $$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 $$y=25x+3 and $$y=120×2−x are examples of the general exponential function given by $$y=A×bmx+c+k, with both $$A and $$m non-zero. The number $$b is known as the base of the function, and it is strictly defined as a positive number not equal to 1. 

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

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

The constant $$m is often called the growth constant (or decay constant if $$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 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
$$y=−3x	Reflect $$y=3x across the $$x-axis
$$y=3x−5	Translate $$y=3x horizontally  to the right by $$5 units
$$y=3x−5	Translate $$y=3x vertically downward by $$5 units
$$y=2×3x	Double every $$y value of  $$y=3x
$$y=8−3x	Reflect $$y=3x across the $$x axis then translate $$8 units upward

More complex forms of the exponential require more thought. For example, the function $$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=32(x−52​)=(32)(x−52​)=9(x−52​).

Hence, the function could be thought of as the function $$y=9x translated to the right by $$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 $$y-intercepts of each graph, the limiting value of $$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	$$y-intercept	limiting value	rising/falling
$$y=2x	$$y=1	$$y=0	rising
$$y=3−x+1	$$y=1	$$y=0	falling
$$y=3×4x−2	$$y=1	$$y=−2	rising
$$y=(0.5)x	$$y=1	$$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 $$y=(0.5)x is none other than $$y=2−x in disguise. Thus: 

$$y=(0.5)x=(12​)x=12x​=2−x 

 

In a similar way we can say that $$y=(1b​)x=b−x, and so every exponential curve of the form $$y=bx, with a base $$b in the interval $$0<b<1, can be re-expressed as $$y=(1b​)−x. Since $$b is a positive number, this means that exponential functions of the form  $$y=bx where $$0<b<1 are in fact decreasing curves.  

Worked Examples

Question 1

Question 2

Question 3