Download 4.1.2: Properties of Exponential Functions

Introduction
An exponential function has various key features, or
characteristics. Two of these are the domain, which is the
set of all input values (x-values) that satisfy the given
function without restriction, and the the set of all outputs
(y-values) that are valid for the function. Other key
features include the y-intercept(s), asymptote(s), and end
behavior. The y-intercept is the point at which the graph
of a function crosses the y-axis, written (0, y). The
asymptote is an equation that represents a set of points
that are not allowed by the conditions in a parent function
or model; shown on a graph, the asymptote is a line that a
function gets closer and closer to, but never touches.
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4.1.2: Properties of Exponential Functions
Introduction, continued
The end behavior is the behavior of the graph as x
approaches positive or negative infinity, and can be
described as whether the function’s graph increases or
decreases within its domain. These key features can be
determined by analyzing the values of a, b, and c in the
general form of the equation of the function, or by viewing
a graph of the function.
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4.1.2: Properties of Exponential Functions
Key Concepts
• Recall that the general form of an exponential function
is f(x) = a(bx) + c, where a, b, and c are constants and b
is greater than 0 but not equal to 1.
• If a = 1 and c = 0, f(x) = a(bx) + c can be rewritten as
f(x) = bx.
• The exponential function f(x) = bx has a domain of all
real numbers, a range of all real numbers greater than
0, a y-intercept of 1, and an asymptote that is the x-axis.
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4.1.2: Properties of Exponential Functions
Key Concepts, continued
• In general, if a is positive, a function of the form f(x) =
a(bx) + c has a domain of all real numbers, a range of
all real numbers greater than c, a y-intercept of a + c,
and an asymptote of y = c.
• If b > 1, the function increases within its domain, and if
0 < b < 1, the function decreases within its domain. In
other words, the function shows exponential growth if
b > 1 and exponential decay if 0 < b < 1.
• The domain and range of an exponential function may
be restricted if the function is used to model a realworld scenario.
4.1.2: Properties of Exponential Functions
4
Key Concepts, continued
•
The graph of an exponential function can be obtained
by using a graphing calculator.
On a TI-83/84:
Step 1: Press [MODE].
Step 2: Make sure Func is highlighted on the fourth line.
If not, arrow down to it and press [ENTER].
Step 3: Press [Y=]. Press [CLEAR] to delete any
equations.
Step 4: Enter the equation of the exponential function at
Y1. Use [X, T, θ, n] for variables and [^] for
exponents.
(continued)
4.1.2: Properties of Exponential Functions
5
Key Concepts, continued
Step 5: If needed, press [WINDOW] to adjust the viewing
window. Change the settings as appropriate.
Step 6: Press [GRAPH].
On a TI-Nspire:
Step 1: Press [home]. Arrow over to the graphing icon
and press [enter].
Step 2: Press [menu]. Use the arrow key to select 3:
Graph Type and then 1: Function. Press [enter].
Step 3: Use your keypad to enter the equation of the
exponential function after f1(x).
(continued)
4.1.2: Properties of Exponential Functions
6
Key Concepts, continued
Step 4: If needed, adjust the viewing window. Press
[menu], then select 4: Window/Zoom, and then
1: Window Settings. Change the settings as
appropriate.
Step 5: Press [enter].
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4.1.2: Properties of Exponential Functions
Common Errors/Misconceptions
• not taking into account the values of a and c in an
exponential function of the form f(x) = a(bx) + c when
determining the function’s range, y-intercept, and/or
asymptote
• not restricting the domain and range of an exponential
function given a real-world scenario
• forgetting to switch a graphing calculator to function
mode
• setting a graphing calculator’s window variables
inappropriately
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4.1.2: Properties of Exponential Functions
Guided Practice
Example 1
What are the domain, range, y-intercept, asymptote, and
end behavior of the exponential function f(x) = 7x? Confirm
your results by graphing.
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4.1.2: Properties of Exponential Functions
Guided Practice: Example 1, continued
1. Determine the values of a, b, and c in the
function.
The function is in the form f(x) = a(bx) + c, so a = 1,
b = 7, and c = 0.
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4.1.2: Properties of Exponential Functions
Guided Practice: Example 1, continued
2. Use the values of a and c to find the
domain, range, y-intercept, and asymptote
of the function.
From the previous step, it is known that a = 1 and c = 0.
Recall that if a is positive, a function of the form
f(x) = a(bx) + c has a domain of all real numbers, a
range of all real numbers greater than c, a y-intercept
of a + c, and an asymptote of y = c.
Since a is positive, the domain is all real numbers.
The range is all real numbers greater than c, which is 0.
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4.1.2: Properties of Exponential Functions
Guided Practice: Example 1, continued
The y-intercept is a + c, or 1 + 0, which simplifies to 1.
The asymptote is y = c, or y = 0.
Therefore, the domain of f(x) = 7x is all real numbers,
the range is all real numbers greater than 0, the
y-intercept is 1, and the asymptote is the x-axis (y = 0).
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4.1.2: Properties of Exponential Functions
Guided Practice: Example 1, continued
3. Use the value of b to determine the
function’s end behavior.
The value of b in the function f(x) = 7x is 7, and 7 > 1.
When any number greater than 1 is raised to a larger
and larger power, the resulting value also becomes
larger and larger. Therefore, the function increases
within its domain.
4.1.2: Properties of Exponential Functions
Guided Practice: Example 1, continued
4. Graph the function and use it to confirm
your findings.
Graph the function f(x) = 7x using the directions
appropriate to your calculator model. Compare the key
features of the graph (domain, range, y-intercept,
asymptote, and end behavior) with your findings.
On a TI-83/84:
Step 1: Press [MODE].
Step 2: Make sure Func is highlighted on the fourth line.
If not, arrow down to it and press [ENTER].
(continued)
4.1.2: Properties of Exponential Functions
Guided Practice: Example 1, continued
Step 3: Press [Y=]. Press [CLEAR] to delete any
equations.
Step 4: Enter the equation of the exponential function at
Y1. Use [X, T, θ, n] for variables and [^] for
exponents.
Step 5: If needed, press [WINDOW] to adjust the
viewing window. Change the settings as
appropriate.
Step 6: Press [GRAPH].
(continued)
4.1.2: Properties of Exponential Functions
Guided Practice: Example 1, continued
On a TI-Nspire:
Step 1: Press [home]. Arrow over to the graphing icon
and press [enter].
Step 2: Press [menu]. Use the arrow key to select 3:
Graph Type and then 1: Function. Press [enter].
Step 3: Use your keypad to enter the equation of the
exponential function after f1(x).
Step 4: If needed, adjust the viewing window. Press
[menu], then select 4: Window/Zoom, and then
1: Window Settings. Change the settings as
appropriate.
Step 5: Press [enter].
4.1.2: Properties of Exponential Functions
Guided Practice: Example 1, continued
The graph extends
infinitely in both
directions, so the
domain is all real
numbers.
The graph includes
only y-values
greater than 0,
which means the
range is all real
numbers greater
than 0.
4.1.2: Properties of Exponential Functions
Guided Practice: Example 1, continued
The graph intersects the y-axis at (0, 1), so the
y-intercept is 1.
The graph approaches but never touches the x-axis;
therefore, the asymptote is the x-axis (y = 0).
This graph shows the function values rising from left
to right, so the function is increasing within its domain.
Thus, the graph confirms the findings from the
previous steps.
✔
4.1.2: Properties of Exponential Functions
Guided Practice: Example 1, continued
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4.1.2: Properties of Exponential Functions
Guided Practice
Example 2
What are the domain, range, y-intercept, asymptote, and
x
end behavior of the exponential function g(x) = 6
æ1æ
?
+
5
æ
æ8 æ
æ
Confirm your results by graphing.
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4.1.2: Properties of Exponential Functions
Guided Practice: Example 2, continued
1. Determine the values of a, b, and c in the
function.
The function is in the form f(x) = a(bx) + c, so a = 6,
1
b = , and c = 5.
8
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4.1.2: Properties of Exponential Functions
Guided Practice: Example 2, continued
2. Use the values of a and c to find the
domain, range, y-intercept, and asymptote
of the function.
From the previous step, it is known that a = 6 and
c = 5.
Recall that if a is positive, a function of the form f(x) =
a(bx) + c has a domain of all real numbers, a range of
all real numbers greater than c, a y-intercept of a + c,
and an asymptote of y = c.
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4.1.2: Properties of Exponential Functions
Guided Practice: Example 2, continued
Since a is positive, the function follows this rule.
Therefore, the domain is all real numbers, the range
is all real numbers greater than 5, the y-intercept is
6 + 5 = 11, and the asymptote is y = 5.
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4.1.2: Properties of Exponential Functions
Guided Practice: Example 2, continued
3. Use the value of b to determine the
function’s end behavior.
The value of b is
1
, and 0 <
1
< 1 . When any number
8
8
greater than 0 and less than 1 is raised to a larger and
larger power, the resulting values become smaller and
smaller. Therefore, the function decreases within its
domain.
4.1.2: Properties of Exponential Functions
Guided Practice: Example 2, continued
4. Graph the function and use it to confirm
your findings.
x
æ1æ
Graph the function g(x) = 6 æ æ + 5 on a graphing
æ8 æ
calculator.
4.1.2: Properties of Exponential Functions
Guided Practice: Example 2, continued
The graph
confirms that the
domain is all real
numbers, the
range is all real
numbers greater
than 5, the
y-intercept is 11,
and the asymptote
is y = 5.
✔
4.1.2: Properties of Exponential Functions
Guided Practice: Example 2, continued
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4.1.2: Properties of Exponential Functions