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.
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 $$m to non-zero integer values only.
We introduce all of these constants in order to accurately model real world phenomena. This is the power of a generalised model. We can adjust the constants to fit reality and in so doing learn more about the way nature works.
Whilst the general form is a comprehensive tool for sketching exponential curves, there are a few simpler observations to keep in mind. We can summarise them using examples as shown in this table:
Specific Example | Observation |
---|---|
$$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 $$212 units.
The applet below is extremely versatile, but we need to keep in mind that it is a learning tool exploring the effects of the different constants involved. As a guide, it might be helpful to use the applet to create the four graphs shown in this table.
Verify the $$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?
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 $$0<b<1 are in fact decreasing curves.
The function $$y=5x has been transformed into the function $$y=5x+4+2
Identify the:
Horizontal translation
Vertical translation
Growth constant
Consider the function $$y=10−x and its inverse function:
Plot the graph of $$y=10−x.
Find the equation of the inverse function of $$y=10−x.
Hence determine the correct graph for $$y=−logx.
Consider the function $$y=3−x :
Find the $$y-intercept of the curve $$y=3−x.
Fill in the table of values for $$y=3−x.
$$x | $$−3 | $$−2 | $$−1 | $$0 | $$1 | $$2 | $$3 |
$$y | $$ | $$ | $$ | $$ | $$ | $$ | $$ |
Find the horizontal asymptote of the curve $$y=3−x.
Hence plot the curve $$y=3−x.
Is the function $$y=3−x, an increasing or decreasing function?
Increasing function
Decreasing function