7.1 Exponential Functions
x
 1
Exponential functions, such as y = 3x and y =   , are used to model a wide variety of
 4
natural phenomena. For example, a bacterial culture that doubles in size every hour might
be modelled by the function y = 2t, where t is in hours. The amount of a radioactive isotope
t
 1  1000
with a half-life of 1000 years might be modelled by the function y =  
, where t is in
 2
years.
Exponential functions can be written in the form f(x) = ax, where the base, a, is a
positive constant not equal to 1, that is, a Î (0, 1) or a Î (1, ¥). The exponent, x, is a
variable that can be any real number, unless there is some restriction on the domain.
Exponential relationships are used to model compound interest, population growth, and
resource consumption.
We can perform all the usual transformations—translations, reflections, and stretches—on
the graphs of exponential functions. The following investigation explores the effect of these
transformations on exponential functions.
Investigate & Inquire: Transforming Exponential
Functions
x
1. Graph the function y = 2 . Then, graph each function below and use the information
from the graphs to copy and complete the table.
Equation
Transformation
of graph of y = 2 x
Domain
End
Range Behaviour
Equation of
asymptote(s)
Intercepts
x
y=2 +3
y = 2x - 5
x
y = (3)2
y = (-4)2x
y = (-4)2x + 3
x
x
2. a) Describe the effect of c on the graph when transforming y = a into y = ca . What
happens to the graph if c < 0?
x
x
b) Describe the effect of b on the graph when transforming y = a into y = a + b
i) if b > 0
ii) if b < 0
3.
a)
b)
c)
Without constructing a table of values,
x
x
describe how to graph y = (2)3 + 3, given the graph of y = 3
x
x
describe how to graph y = (-3)5 - 7, given the graph of y = 5
x
x
describe how to graph y = ba + c, given the graph of y = a
Next, we will explore changing the value of the base, a.
414
MHR Chapter 7
Investigate & Inquire: How the Base of an Exponential
Function Influences Its Graph
1. Use graphing technology to investigate the graphs of exponential functions with
different bases.
x
x
x
x
a) Graph y = 2 , y = 3 , y = 5 , and y = 11 .
x
x
x
x
 1
 1
 1
 1
b) Graph y =   , y =   , y =   , and y = 
.

 2
 3
 11 
 5
x
2. a) Describe the changes to the graphs of y = a as a increases, for a Î (1, ¥).
x
b) Describe the changes to the graphs of y = a as a decreases, for a Î (0, 1).
3. a) Compare the graphs in step 1, parts a) and b), in pairs. That is, compare the graphs
x
x
1
 1
of y = 2x and y =   , and then the graphs of y = 3x and y =   , and so on. How are
 3
 2
the graphs in each pair alike? How are they different?
x
x
 1
b) Without graphing, describe how the graphs of y = 6 and y =   differ and what they
 6
have in common. Use graphing technology to confirm your answer.
4. What point do all exponential functions appear to have in common? Explain why this is so.
x
x
5. a) Graph the functions y = 0 and y = 1 . Describe the graphs. Are these exponential
functions? Explain.
x
b) What happens if you try to graph y = (-2) on a graphing calculator or graphing
x
software? Set up a table of values of y = (-2) using the TABLE SETUP screen, with a
TblStart value of -2 and a ∆Tbl value of 0.1. Explain why we have the restriction
a Î (0, 1) or a Î (1, ¥) for the exponential function y = ax.
The graph of f (x) = ax, where a Î (1, ¥) and
x Î (-¥, ¥), is continuous and always increasing.
When a Î (0, 1), the graph is continuous and
always decreasing.
Look at the graphs of the exponential functions
x
f(x) = a for various values of the base a. Notice that,
regardless of the base, all of these graphs pass through
the same point, (0, 1), because a0 = 1 for a ¹ 0. Also,
note that the x-axis is a horizontal asymptote and
that the graphs never touch or cross the x-axis,
since ax > 0 for all values of x. Thus, the
exponential function f (x) = ax has domain (-¥, ¥) and
range (0, ¥) for a > 0.
xy
y= 1
x
3
y=3
8
x
((
((
y= 2
5
x
((
y= 5
2
6
4
x
((
y= 1
2
–6
–4
x
2
–2
0
y=2
2
4
6
x
Now, we take a closer look at the function y = ax, where a Î (0, 1) or a Î (1, ¥).
a ∈ (1, ¥): As x approaches infinity, the graph of y = ax increases rapidly, and as x
approaches negative infinity, the graph is asymptotic to the x-axis. Thus:
If a Î (1, ¥), lim a x = 0 and lim a x = ¥.
x ® -¥
x®¥
7.1 Exponential Functions MHR
415
Furthermore, the larger the base a, the more rapidly the function increases as x approaches
infinity. As x approaches negative infinity, the larger the base a, the more rapidly the graph
of the function approaches its asymptote, the x-axis.
y
x
((
y= 1
5
x
y=5
y
8
8
x
y=2
6
6
((
y= 1
2
4
–2
0
4
2
2
(0, 1)
–4
x
(0, 1)
2
4
x
–4
–2
0
2
4
x
a ∈ (0, 1): As x approaches infinity, the graph is asymptotic to the x-axis, and as
x approaches negative infinity, the graph of y = ax increases rapidly. Thus:
If a Î (0, 1), lim a x = ¥ and lim a x = 0 .
x ® -¥
x®¥
Furthermore, the smaller the base a, the more rapidly the graph of the function approaches
its asymptote, the x-axis, as x approaches infinity. As x approaches negative infinity, the
smaller the base a, the more rapidly the function increases.
Can a be less than 0 for y = ax? That is, can y = ax have a negative base? The answer is yes,
but the resulting function is so badly discontinuous that it has no practical use. In the
Investigation above, y = (-2)x was graphed. Note from the table of values that, for
1
many values of x, the function is not defined. For example, (−2)2 , or
1
4
−2 , is not
1
6
defined in the real numbers. Neither are (−2) , (−2) , and so on. Since many expressions
with negative bases cannot be evaluated, we restrict the definition of the exponential
function to positive values of a. Since 1x = 1 for all values of x, f(x) = 1x is not considered
an exponential function. Since
 0, x > 0
0x = 
 undefined, x ≤ 0
f (x) = 0x is not considered an exponential function either. Thus, we restrict a in the
exponential function y = ax to positive real numbers not equal to 1, that is, a Î (0, 1)
or a Î (1, ¥).
Example 1 Transformations of an Exponential Function
Use the graph of y = 3x to sketch the graph of each function. Use technology to confirm
your results.
x
a) y = 3 + 4
x
x
b) y = -3 (Note: this is not the same as y = (-3) , which is a discontinuous function.)
416
MHR Chapter 7
Solution
x
x
a) The graph of y = 3 + 4 is obtained by starting with the graph of y = 3 and translating it
four units upward. We see from the graph that the line y = 4 is a horizontal asymptote.
The asymptote has moved up 4 units from the original asymptote, y = 0.
y
y
12
6
10
4
y=3
8
x
2
6
–6
x
y=3 +4
–4
–2
4
2
–4
–2
0
0
2
4
6
x
–2
y=3
2
–4
x
x
4
–6
x
y=–3
x
b) Again, we start with the graph of y = 3 and reflect it in the x-axis to get the graph of
x
y = -3 . The horizontal asymptote is y = 0.
Example 2 Transformations of an Exponential Function
x
x-2
a) Use the graph of y = 2 to sketch the graph of y = 2
- 3.
b) State the asymptote, the domain, and the range of this function.
Solution
x-2
a) To graph y = 2
using the graph of y = 2x, we translate
the original graph 2 units to the right. Then, we graph
y = 2x - 2 - 3 by translating y = 2x - 2 downward 3 units. To
summarize, we graph y = 2x - 2 - 3 by translating the graph
x
of y = 2 to the right 2 units and downward 3 units.
b) We see from the graph that the horizontal asymptote is
y = -3. It has been translated downward 3 units along
with the graph.
The domain is R and the range is (-3, ¥).
y
8
6
4
2
–4
–2
0
–2
y=2
x
2
4
y=2
(x – 2)
6
x
–3
Example 3 An Exponential Model for Light Transmitted by Water
The equation s = 0.8d models the fraction of sunlight, s, that reaches a scuba diver under
water, where d is the depth of the diver, in metres.
d
a) Use graphing technology to graph the sunlight, s = 0.8 , that reaches a scuba diver, using
a suitable domain and range.
b) Determine what percent of sunlight reaches a diver 3 m below the surface of the water.
7.1 Exponential Functions MHR
417
Solution
d
a) The graph of s = 0.8 is shown. For Window variables, we use
the domain [0, 20] because depth cannot be negative. We use trial
and error to find a suitable upper limit of the domain. Since s is a
proportion, we use the range [0, 1].
b) Use the r key to find the value of s when d = 3.
Confirm the solution algebraically.
s = d3
= 0.83
= 0.512
This means that only about 50% of the sunlight above the
water will reach a diver 3 m below the surface of the water.
Example 4 Limit of an Exponential Function
1
Find lim- 4 x - 2 .
x®2
Solution 1 Graphing Calculator Method
1
Since the function f (x) = 4 x − 2 is not continuous at x = 2, we cannot substitute x = 2 to find
the limit. We will use a table of values to determine the limit. Enter the function in the
Y= editor of a graphing calculator. Then, set up a table using the TABLE SETUP screen.
Use Ask mode for the independent variable (x, in this case).
1
Enter x-values that approach 2 from the left. From the table, it appears that 4 x − 2
approaches zero very quickly as x approaches 2 from the left. Thus, it appears that
1
lim− 4 x − 2 = 0 .
x→2
Solution 2 Paper and Pencil Method
1
x−2
1
as lim 4z , where z =
. Recall
z ® -¥
x−2
As x approaches 2 from the left, x - 2 approaches zero, and is negative. Thus,
1
approaches negative infinity. We can rewrite lim− 4 x − 2
x→2
1
from earlier that, if a Î (1, ¥), then lim a x = 0 . Thus, lim 4z = 0 , so lim− 4 x − 2 = 0 .
x ® -¥
418
MHR Chapter 7
z ® -¥
x→2
Key Concepts
·
·
·
·
If a Î (1, ¥), then lim a x = 0 and lim a x = ¥. The function f (x) = ax has domain
x ® -¥
x®¥
x Î (-¥, ¥) and range y Î (0, ¥).
If a Î (0, 1), then lim a x = ¥ and lim a x = 0 . The function f (x) = ax has domain
x®¥
x ® -¥
x Î (-¥, ¥) and range (0, ¥).
The graph of y = - ax is a reflection of the graph of y = ax in the x-axis.
The graph of y = cax - p + b is obtained by graphing the original function y = ax, and
transforming it as follows:
a) Stretch the graph vertically by a factor of c if |c| > 1; compress the graph
vertically by a factor of c if 0 < |c| < 1.
b) If c < 0, reflect the graph in the x-axis.
c) Translate the graph left p units if p > 0, and right p units if p < 0.
d) Translate the graph up b units if b > 0, and down b units if b < 0.
Communicate Your Understanding
1. a) Is the domain of every exponential function the same? Explain.
b) Is the range of every exponential function the same? Explain.
x
2. a) Given the graph of y = 3 , explain how to graph y = æç
x
technology or a table of values.
1 ö without using
÷
è 3ø
x
x
b) Given the graph of y = 6 , describe how to graph y = (-4)6 - 5.
x
3. Explain why the function y = a is not considered an exponential function
a) when a = 1
b) when a = 0
c) when a < 0
Practise
and differences
A 1. a) Explain the similarities
x
x
x
among the graphs of y = 2 , y = 6 , and y = 9 .
In your explanation, pay attention to the
y-intercepts and the limits as x approaches
x
positive and negative infinity.
æ1ö
y
=
b) Repeat part a) for the graphs of
ç2÷ ,
è ø
x
x
æ1ö
æ 1ö
y = ç ÷ , and y = ç ÷ .
è9ø
è6ø
2. Graph each pair of functions on the same set
of axes. First, use a table of values to graph f (x),
and then, use your graph of f (x) to graph g (x).
Check your work using graphing technology.
x
x
æ1ö
a) f (x) = 3 and g(x) = ç ÷
è 3ø
x
æ1ö
÷
è5ø
x
æ 1ö
x
c) f (x) = 6 and g(x) = ç ÷
è6ø
x
x
æ 1 ö
d) f (x) = 10 and g(x) = ç
÷
è 10 ø
b) f (x) = 5 and g(x) = ç
x
x
3. a) Use a table of values to graph f (x) = 2 .
æ 1ö x
÷ (2 ) and
è 3ø
h(x) = (3)2x using the graph of f (x) = 2x.
æ1ö x
x
c) Graph g(x) = ç ÷ (2 ) and h(x) = (3)2 on
è 3ø
the same set of axes as f (x). Confirm your results
with graphing technology.
d) State the domain and range of each function.
b) Describe how to graph g(x) = ç
7.1 Exponential Functions MHR
419
4. Use the given graphs to sketch a graph of
each of the following functions without using a
table of values or technology. State the
y-intercept, domain, range, and equation of the
asymptote of each function.
y
1
e) f (x) = −3  
 4
c)
2
–2
x
x ® -¥
4
x
y=2
0
y
2
e)
4
x
g)
i)
6
x
b) g(x) = -4
c) h(x) = (-2)7 - 5
6. Evaluate.
x
a) lim 3
6
–4
x
a) f (x) = 3 + 2
lim 4- x
x®¥
lim- 3
1
x
x®0
lim 6
1
x
x
lim (2 + 1)
x
x
x
f)
b)
 1
h(x) = 2   − 7
 2
lim 83 x -1
x®¥
d) lim 2 x
x®¥
1
f)
h)
x®¥
x ® -¥
 1
d) g(x) =   + 1
 3
lim+ 3 x
x→0
lim 5- x
2
x®¥
1
j)
lim− 3 x − 4
x→4
4
–4
–2
Apply, Solve, Communicate
x
2
y=3
0
y
8
B 7. Application In Example 3 on page 417, it
would be almost completely dark underwater if
only 0.1% of the sunlight above the water
reached a diver. At what depth does this
occur?
4 x
2
6
4
–4
–2
x
2
y=4
0
2
4
x+2
x
a) f (x) = -2
x
c) h(x) = 2 + 3
e)
 1
y= − 
 3
x
b) g(x) = 3
x
d) f (x) = (3)3 - 5
x
x
f)
 1
 3 
1
f (x) =   + 3
 2
x+4
g) h(x) = (−2) 
−6
x+3
h)
 1
y = (3)  
 4
5.
i)
ii)
iii)
iv)
For each function,
state the y-intercept
state the domain and range
state the equation of the asymptote
draw a graph of the function
420
+2
MHR Chapter 7
8. Inquiry/Problem Solving Randolf bought
a computer system for $4000. The
system depreciates at a rate of 20%
each year.
a) Determine an exponential function to model
the value of the system over time.
b) Graph the function you found in
part a).
c) Use your graph to determine the value
of Randolf’s computer system 3 years from
now.
d) Verify your result from part c)
algebraically.
9. Inquiry/Problem Solving Roy bought an
antique slide rule in 2001 for $25. The value of
the collector’s item is increasing and can be
approximately determined by the expression
y = 25(1.03)t, where t is the number of
years since 2001, and y is the value,
in dollars.
a) Graph the value of the slide rule over
time.
b) Find the approximate value of the slide rule
d) Does the result in part c) make sense?
in 2015.
c) Determine approximately when the slide rule
will have a value of $50.
Comment on the domain of validity of the model.
e) According to nuclear physics, the
transformation of radioactive atoms is a discrete
process, that is, every so often an individual
atom transforms. There is always a whole
number of untransformed atoms remaining.
Thus, a graph of this process would not be
continuous. Yet the exponential model is
continuous. Comment on the validity of using
a continuous model for a phenomenon that is
essentially discrete. Explain why the model is
nevertheless a good one. Over what domain is
the model good?
10. Collette bought a $1500 compound
interest savings bond with a 5% annual
interest rate.
a) Graph the growth of Collette’s savings bond
over time.
b) Use your graph to determine the value of her
bond when it matures 15 years from now.
Web Connection
To find current and historical values for the
Bank of Canada prime lending rate, go to
www.mcgrawhill.ca/links/CAF12 and follow the
link.
C 13. Explain how you would graphx the following
functions given the graph of y = 2 . Then, graph
each function and state its domain and range.
|x|
-|x|
a) y = 2
b) y = 2
14. Evaluate each limit.
a)
x
11. Application If f (x) = 4 , show that
lim 5 x
b)
lim 3- x
d)
x®¥
lim 5 x
x ® -¥
lim 3- x
f (x + h) − f (x)
 4h − 1  .
= 4x 
 h 
h
c)
12. Communication A sample of radioactive
functions with positive bases are continuous
functions. In this exercise, you will explore
the meaning of exponential expressions with
irrational exponents.
3
a) Explain the meaning of 5 2 in terms of
powers and roots.
1.4
b) Explain the meaning of 5 in terms
of powers and roots.
1.41
c) Explain the meaning of 5
in terms
of powers and roots.
1.414
d) Explain the meaning of 5
in terms
of powers and roots.
1.4142
e) Explain the meaning of 5
in terms of
powers and roots.
f) Use the idea behind parts a) to e), and
the idea of the limit, to explain the meaning
of 5 2 .
iodine-131 atoms has a half-life of about 8 days.
This means that after 8 days, half of the atoms
will have transformed into some other type of
atom. A formula that models the number of
-
t
iodine-131 atoms that remain is P = P0 (2 8 ),
where P is the number of iodine-131 atoms that
remain after time t, in days, and P0 is the
number of iodine-131 atoms that are initially
present. Suppose that 1000000 iodine-131
atoms are initially present.
a) How many iodine-131 atoms remain after
24 days?
b) How many iodine-131 atoms remain after
80 days?
c) How many iodine-131 atoms remain after
360 days?
x®¥
x ® -¥
15. In this section, we stated that exponential
7.1 Exponential Functions MHR
421