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.
General shape and position of the graph of a function of the form
.
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
.
Economics is important because it helps people understand how a variety of factors work with and against each other to control how resources such as labor and capital get used, and how inflation, supply, demand, interest rates and other factors determine how much you pay for goods and services.
Muhammad
explain the steps taken by the government in developing rural market?
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Soumya
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Mitiku
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Soumya
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Mitiku
parna kya hai behencho?
Soumya
Hallo
Rabindranath
Hello, dear what's up?
Mitiku
cool
Momoh
good morning
Isaac
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Isaac
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Afran
Ys please I'm in Ghana
Afran
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OLANIYI
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OLANIYI
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OLANIYI
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OLANIYI
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akinwale
am from Ghana
Jacob
Pls economic A level exam tomorrow pls help me
akinwale
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Dev
bol Diya discuss
ab krega v
Dev
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Dev
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Dev
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OLANIYI
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JOSEPH
I want to know about economic A level tomorrow pls help
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.