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Chapter 8, Sections 1 & 2
Exponential Functions
Exponential Functions
Exponential functions are functions where the
variable appears in the exponent, for
example:
y = 2x; or…
y = baseexponent
The general form of an exponential
function is y = abx, where a ≠ 0, and
b > 0 and b ≠ 1.
Characteristics of an
Exponential Function
1. It is continuous and one-to-one
2. The domain is all real numbers
3. The x-axis is an asymptote of the graph
asymptote: line that the graph approaches as you
move away from the origin
4. The range is all positives when a > 0,
all negatives when a < 0.
5. The graph contains (0,a); a is the y-intercept.
(Why?)
6. The graphs of y = abx and y = a(1/b)x are
reflections of each other across the y-axis.
Example 1
Sketch the graph of y = 4x. State
the function’s domain and
range.
We can make a table of values to
get the general idea:
x
y
0
1
1
4
2
16
3
64
-1
¼
-2
1/16
-3
1/64
We see the domain is all reals, and
the range is all positives.
Exponential growth and decay
What does the function y = 1x look like?
This is NO change -- no increase, no
decrease.
When the BASE is GREATER THAN 1, we call
this exponential growth.
When the BASE is LESS THAN 1 (but greater
than zero) we call it exponential decay.
Example 1
Determine whether each function represents
exponential GROWTH or DECAY:
y = (0.7)x
Exponential DECAY
y = ½(3)x
Exponential GROWTH
y = 10(4/3)x
Exponential GROWTH
y = 8(3/2)x
Exponential GROWTH
y = 10(3)-x
Exponential DECAY
Exponential Growth and Decay Model y = abx
Let a and b be real numbers, with a > 0 ,
If b > 1, the model
is exponential growth
If 0 < b < 1, the model
is exponential decay
Exponential Growth and Decay
Graph, identify if growth or decay, give the
domain and range
1. y = ½ • 3x
2. y = 5(2/3)x
3. y = -(3/2)x
4. y = -2 • 2x
5. y = 8 •½x
Writing Exponential Functions
When we assume a constant rate of change
(growth or decay), this can be modeled by an
exponential function.
Exponential Growth Model: y = a(1 + r)t
1 + r is the growth factor
Exponential Decay Model: y = a(1 – r)t
1 – r is the decay factor
a = initial amount
r = rate (decimal)
t = time
Examples
In January, 1993, there were about 1,313,000
Internet hosts. During the next five years,
the number of hosts increased by 100% per
year.
a. Write an exponential model giving the
number h of hosts t years after 1993.
b. About how many hosts were there in 1996?
You buy a new car for $24,000. The value of
the car decreases by 16% per year.
a. Write an exponential model for the value of
the car.
b. What will the car be worth in 2 years?
Try These
Identify if it is a growth/decay, give domain and
range
1. y = 3(1/4)x
2. y = 1/3(2)-x
3. y = 4(3)x
4. y = -2(1/3)x
In 1990 the cost of tuition at a state university was
$4,300. During the next 8 years, the tuition rose 4%
each year. Write an exponential model that gives the
tuition t years after 1990. How much will tuition be
in the year 2010?
A new car costs $23,000. The value decreases
by 15% each year. Write an exponential
model for the car’s value. Use the model to
estimate the value after 3 years.