In Grade 10, you studied graphs of many different forms. In this chapter, you will learn a little more about the graphs of exponential functions.
Functions of the form
For
This form of the exponential function is slightly more complex than the form studied in Grade 10.
Investigation : functions of the form
On the same set of axes, with
and
, plot the following graphs:
Use your results to understand what happens when you change the value of
.
You should find that the value of
affects whether the graph curves upwards (
) or curves downwards (
). You should also find that a larger value of
(when
is positive) stretches the graph upwards. However, when
is negative, a lower value of
(such as -2 instead of -1) stretches the graph downwards. Finally, note that when
the graph is simply a horizontal line. This is why we set
in the original definition of these functions.
On the same set of axes, with
and
, plot the following graphs:
Use your results to understand what happens when you change the value of
.
You should find that when
is increased, the whole graph is translated (moved) upwards. When
is decreased (poosibly even made negative), the graph is translated downwards.
On the same set of axes, with
and
, plot the following graphs:
Use your results to understand what happens when you change the value of
.
You should find that the value of
affects whether the graph curves upwards (
) or curves downwards (
). You should also find that a larger value of
(when
is positive) stretches the graph upwards. However, when
is negative, a lower value of
(such as -2 instead of -1) stretches the graph downwards. Finally, note that when
the graph is simply a horizontal line. This is why we set
in the original definition of these functions.
Following the general method of the above activities, choose your own values of
and
to plot 5 graphs of
on the same set of axes (choose your own limits for
and
carefully). Make sure that you use the same values of
,
and
for each graph, and different values of
. Use your results to understand the effect of changing the value of
.
These different properties are summarised in
[link] .
Table summarising general shapes and positions of functions of the form
.
Domain and range
For
, the function is defined for all real values of
. Therefore, the domain is
.
The range of
is dependent on the sign of
.
If
then:
Therefore, if
, then the range is
. In other words
can be any real number greater than
.