10-5 Exponential Functions
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Objectives
• Students will graph exponential
functions.
• Students will identify data that displays
exponential behavior.
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Exponential Function
• An exponential function is a function
that can be described by an equation of
the form y = ax, where a>0 and a≠
≠1.
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Graph
State the y-intercept.
x
3x
–1
3–1
0
30
1
1
31
3
2
32
9
y
Graph the ordered pairs
and connect the points with
a smooth curve.
Answer: The y-intercept is 1.
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Use the graph to determine the approximate
value of
The graph represents all real
values of x and their corresponding
values of y for
Answer: The value of y is about
5 when
Use a calculator to confirm
this value.
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a. Graph
State the y-intercept.
Answer: The y-intercept is 1.
b. Use the graph
to determine the
approximate value
of
Answer: about 1.5
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Graph
State the y-intercept.
x
y
–1
4
0
1
Graph the
ordered pairs
and connect the
points with a
smooth curve.
1
Answer: The y-intercept is 1.
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Use the graph to determine the
approximate value of
Answer: The value of y is about
8 when
Use a calculator to confirm
this value.
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a. Graph
State the y-intercept.
Answer: The y-intercept is 1.
b. Use the graph to
determine the approximate
value of
Answer: about 3
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Depreciation People joke that the value of a new car
decreases as soon as it is driven off the dealer’s lot.
The function
models the
depreciation of the value of a new car that originally
cost $25,000. V represents the value of the car and t
represents the time in years from the time the car was
purchased. Graph the function. What values of V and t
are meaningful in the function.
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10
Use a graphing calculator to graph the function.
Answer:
Only the values of V ≤ 25,000 and t ≥ 0 are meaningful
in the context of the problem.
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11
What is the value of the car after one year?
Original equation
Use a calculator.
Answer: After one year, the car’s value is about $20,500.
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12
What is the value of the car after five years?
Original equation
Use a calculator.
Answer: After five years, the car’s value is about $9270.
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13
Depreciation The function
models
the depreciation of the value of a new car that
originally cost $22,000. V represents the value of the
car and t represents the time in years from the time
the car was purchased.
a. Graph the function.
What values of V
and t are meaningful
in the function.
Answer: Only the values
of V ≤ 22,000 and t ≥ 0
are meaningful in the
context of the problem.
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14
b. What is the value of the car after one year?
Answer: $18,040
c. What is the value of the car after three years?
Answer: $12,130
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15
Determine whether the set of data displays
exponential behavior.
x
y
0
10
10
25
20
62.5
30
156.25
Method 1 Look for a Pattern
The domain values are at regular intervals of 10.
Look for a common factor among the range values.
10
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62.5
156.25
16
16
Answer: Since the domain values are at regular intervals
and the range values have a common factor, the data are
probably exponential. The equation for the data may
involve (2.5)x.
Method 2 Graph the Data
Answer: The graph shows a
rapidly increasing value of y
as x increases. This is a
characteristic of exponential
behavior.
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17
Determine whether the set of data displays
exponential behavior.
x
y
0
10
10
25
20
40
30
55
Method 1 Look for a Pattern
The domain values are at regular intervals of 10.
The range values have a common difference of 15.
10
25
+15
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+15
55
+15
18
18
Answer: Since the domain values are at regular intervals
and there is a common difference of 15, the data display
linear behavior.
Method 2 Graph the Data
Answer: The graph is a line,
not an exponential function.
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19
Determine whether the set of data displays
exponential behavior.
a.
x
y
0
100
10
50
20
25
30
12.5
Answer: The domain values are at regular intervals and
the range values have a common factor of
so it is
probably exponential. Also the graph shows rapidly
decreasing values of y as x increases.
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20
Determine whether the set of data displays
exponential behavior.
b.
x
y
0
–5
10
0
20
5
30
10
Answer: The domain values are at regular intervals and
the range values have a common difference of 5. The data
display linear behavior.
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21