Graph Exponential Functions; Graph Using Transformations
Graph f ( x ) = 2 3x+1 − 4 using transformations and the guided visualization app:
https://media.pearsoncmg.com/cmg/pmmg_mml_shared/precalc_ifigs_HTML5/IFig1_29Precalc_graphs_of_exponential_functions/index.html
Graph
What was the
transformation
to the graph?
What
happened?
3𝑥
n/a
3𝑥+1
2 ∙ 3𝑥+1
2 ∙ 3𝑥+1 − 4
Give screenshots of the results for each step.
Equation of
asymptote
Domain
Range
Section 5.3 & 5.4
Chapter 5: Exponential and Logarithmic Functions
Section 5.3: Exponential Functions
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Author in Action: Evaluate Exponential Functions (17:33)
Before delving into exponential functions, let’s make sure we can use our calculators to evaluate
exponential expressions.
Most calculators have either an x y key or a carot key ^ for working with exponents. To evaluate
expressions of the form a x , enter the base a, then press the x y key (or ^ ), enter the exponent x, and
press = , (or enter ).
Post 1
Example 1*: Evaluating Exponential Functions: Using the Calculator
Using a calculator, evaluate:
(a) 21.4
(b) 21.41
(c) 21.414
(d) 21.4142
Laws of Exponents
If s, t, a, and b are real numbers a 0 and b 0 then,
1) a s at = _____
2) (a s )t = _____
4) 1s = _______
5)
a − s = ____ = ____
(e) 2
3)
(ab) s = _____
6)
(a)0 = _____
2
Exploration 1*: Evaluate Exponential Functions
Suppose you are given $10 and told that each day you show up to class the amount you’re given
doubles. Fill in the chart below.
# of Days
you Go to
Class
0
1
2
3
4
5
6
7
8
Pay ($)
10
20
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Exponential and Logarithmic Functions
(a) As the value of the independent variable (days) increases by 1, what is happening to the value of the
dependent variable (pay)?
(b) Create a formula that models this situation.
This situation in Exploration 1 models an exponential growth function.
Definition: An exponential function is a function of the form ______________ where a is a positive
real number (a > 0) and a ≠ 1 and C ≠ 0 is a real number. The domain of f is
_________________________.
The base a is the ______________ factor, and because f ( 0) = Ca0 = C , C is called the
_______________.
End of Post 1
Think/Pair/Share: We will discuss this more in depth later –do you have any thoughts about what
makes a number a growth factor verses a decay factor?
Exploration 2*: Evaluate Exponential Functions: Linear or Exponential?
Now that we have studied both linear and exponential functions, we should be able to look at data and
determine whether it is either of these functions. But how? Let’s explore this.
1. Evaluate f ( x) = 2x and g ( x) = 3x + 2 at x = −2, −1,0,1, 2, and 3
x
f ( x) = 2 x
x
-2
-1
0
1
2
3
g ( x) = 3x + 2
-2
-1
0
1
2
3
2. Comment on the patterns that exist in the values of f and g.
Copyright © 2016 Pearson Education, Inc.
Section 5.3 & 5.4
Theorem: For an exponential function, f ( x) = a x , a 1, a 1 , if x is any real number, then
f ( x + 1)
= _______ or f ( x + 1) = _______
f ( x)
Example 2: Identify Linear or Exponential Functions
Determine whether the given function is linear, exponential, or neither. For those that are linear, find a
linear function that models the data. For those that are exponential, find an exponential function that
models the data.
(a)
(b)
.
x
-1
0
1
2
3
y = f ( x) Average
Rate of
Change
-4.5
-3
-1.5
0
1.5
Ratio of
consecutive
outputs
-1
0
1
2
3
f ( x) = ___________________
(c)
x
y = g ( x) Average
Rate of
Change
-4.5
-3
-1.5
0
1.5
g ( x) = ___________________
(d)
x
-1
0
1
2
3
y = h( x) Average
Rate of
Change
20
16
12
8
4
h( x) = ___________________
Ratio of
consecutive
outputs
Ratio of
consecutive
outputs
.
x
y = j ( x) Average
Rate of
Change
-1
0
1
2
3
2
3
4.5
6.75
10.125
j ( x) = ___________________
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Ratio of
consecutive
outputs
Exponential and Logarithmic Functions
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Author in Action: Graph Exponential Functions (25:55)
Exploration 3*: Graph Exponential Functions
1. Consider the functions f ( x ) = 2x and g ( x ) = 3x . Graph these functions by filling in the table
below. Label each of your graphs.
x
y = f ( x)
x
-2
-3
-1
-2
0
-1
1
0
2
1
y = g ( x)
2
(a) What are the domain and range of each of these functions?
(b) Can y = 0 ? Why or why not?
(c) Does the function have any symmetry?
(d) What are the x and y – intercepts?
(e) Notice that both of these functions are increasing as x increases. What does this mean? Which
function is increasing faster?
(f) Do these functions have a horizontal asymptote?
Copyright © 2016 Pearson Education, Inc.
Section 5.3 & 5.4
Post 2
Properties of the Exponential Function f ( x) = a x , a 1
1. The domain is the set of all real numbers or _______________ using interval notation; the range
is the set of positive real numbers or __________ using interval notation.
2. There are ____ x – intercepts; the y – intercept is ____.
3. The x – axis (_____) is a horizontal asymptote as _______ [ _______________ ].
f ( x) = a x , a 1 is an ____________ function and is _______________.
5. The graph of f contains the points _______, _______, and _______.
6. The figure of f is smooth and continuous with no corners or gaps.
4.
x
x
1
1
2. Now consider the functions f ( x ) = and g ( x ) = . Graph these functions by filling in the
2
3
table below. Label each of your graphs.
x
x
y = f ( x)
y = g ( x)
-2
-2
-1
-1
0
0
1
1
2
2
(a) Why do these functions decrease when the graphs in (1) increased (as x gets larger)?
(b) What are the domain and range of each of these functions?
(c) Do these functions have any asymptotes?
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Exponential and Logarithmic Functions
Properties of the Exponential Function f ( x) = a x , 0 a 1
1. The domain is the set of all real numbers or _______________ using interval notation; the range
is the set of positive real numbers or __________ using interval notation.
2. There are ____ x – intercepts; the y – intercept is ____.
3. The x – axis (_____) is a horizontal asymptote as _______ [ _______________ ].
f ( x) = a x , 0 a 1 is a ____________ function and is _______________.
5. The graph of f contains the points _______, _______, and _______.
6. The figure of f is smooth and continuous with no corners or gaps.
4.
End of Post 2
http://www.mathxl.com/info/exercise.aspx?id=spc10e,5,3,57&se=0
Example 3*: Graph Exponential Functions; Graph Using Transformations
x +1
Graph f ( x ) = 2 3 − 4 and determine the domain, range, and horizontal asymptote of f. Make sure
you graph and label the asymptote(s).
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Section 5.3 & 5.4
As we saw in our Exploration 1 exponential functions are often used in applications involving money.
The act of doubling our money each day says that we are experiencing 100% growth each day. In many
examples involving money, we experience growth on a cycle other than per day. Sometimes, our money
may grow annually, quarterly, or monthly. These different cycles are called different compounding
periods. As we compound more and more often, we say that we are compounding continuously. What is
interesting is that as these compound periods approach ∞, we reach a limit. This limit is the number 𝑒.
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Author in Action: Define the Number e (12:39)
Exploration 4*: Define the Number e
n
1
The number e is defined as e = lim 1 + .
n →
n
Let’s explore this value by filling in this table using a graphing utility:
n
1
f ( n) = 1 +
n
n
10
50
100
500
1000
10,000
100,000
1,000,000
Based on the table, what is e approximately?
Confirm the approximate value of e by typing in e into your calculator
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Exponential and Logarithmic Functions
***We will do more applications with the number 𝑒 in Section 6.7***
Example 4*: Define the Number e; Graph e Using Transformations
Graph f ( x ) = −e x−2 and determine the domain, range, and horizontal asymptote of f.
Solving Exponential Equations
Now that we know what exponential functions are let’s learn about how we can solve exponential
equations. For example, how would you solve the following:
5 x+ 3 =
1
5
What makes this equation different from equations we’ve seen before?
Copyright © 2016 Pearson Education, Inc.
Section 5.3 & 5.4
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Author in Action: Solve Exponential Equations (8:16)
Post 3
Solve Exponential Equations If a u = a v , then __________ .
This means that if you have the same bases on both sides of the equals sign, you set the exponents
equal. The key here is to manipulate as needed so that the base is the same.
Example 5*: Solve Exponential Equations
Solve each equation.
(a)* 23 x -1 = 32
(c) 42 x−5 =
(e) 5x
2
+8
1
16
= 1252 x
(g)* e 2 x −1 =
4
1
e− x )
3x (
e
(b) 5x = 5−6
(d) 22 x−1 = 4
(f) 92 x 27 x = 3−1
2
1
(h)
2
x −5
( )
= (8x ) 2 x
2
End of Post 3
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Exponential and Logarithmic Functions
We will return to applications for the second discussion post.
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Application of Exponential Function (9:14)
Example 6*: Application of Exponential Functions
Between 12:00 PM and 1:00 PM, cars arrive at Citibank’s drive – thru at a rate of 6 cars per hour (0.1
car per minute). The following formula from probability can be used to determine the probability that a
car will arrive within t minutes of 12:00 PM:
F (t ) = 1 − e−0.1t
(a) Determine the probability that a car will arrive within 10 minutes of 12:00 PM (that is, before 12:10
PM).
(b) Determine the probability that a car will arrive within 40 minutes of 12:00 PM (before 12:40 PM).
(c) What values does F approach as t becomes unbounded in the positive direction?
(d) Graph F using a graphing utility.
(e) Using TRACE, determine how many minutes are needed for the probability to reach 50%.
Copyright © 2016 Pearson Education, Inc.
Section 5.3 & 5.4
Section 5.4: Logarithmic Functions
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Author in Action: Convert Exponential to Logarithmic Statements & Logarithmic to Exponential
Statements (11:35)
Post 4
Exploration 1: Logarithms
Before we define a logarithm, let’s play around with them a little. See if you can follow the pattern
below to be able to fill in the missing pieces to a – f.
1
2
log3 9 = 2
log 9 3 =
log 4 16 = 2
log3 27 = 3
(a) log 2 8 = ___
(b) log 4 16 = ___
(c) log ___ 64 = 2
(d) log ___ 64 = 3
(e) log 2 ____ = 4
(f) log 4 2 = ___
Logarithms - A logarithm is just a power
For example, log 2 (32) = 5 says “the logarithm with base 2 of 32 is 5.” It means 2 to the 5th power is 32.
Notice that both in logarithms and exponents, the same number is called the base.
The logarithmic function with base a, where a 0 and a 1 , is denoted by y = log a x (read as “y is
the logarithm to the base a of x”) and is defined by:
________________________________
The domain of the logarithmic function y = logax is ___________.
Example 1*: Convert Exponential to Logarithmic Statements
Change each exponential equation to an equivalent equation involving a logarithm
(a) 58 = t
(b) x −2 = 12
(c) e x = 10
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Exponential and Logarithmic Functions
Example 2*: Convert Logarithmic to Exponential Statements
Change each logarithmic equation to an equivalent equation involving an exponent.
(a) y = log 2 21
(b) log z 12 = 6
(c) log 2 10 = a
End of Post 4
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Author in Action: Evaluate Logarithmic Expressions (4:07)
Example 3*: Evaluate Logarithmic Expressions
Evaluate the following:
1
(b)* log 2
(a)* log3 (81)
(c) log5 (1)
8
(e) log3 (9)
(f) log 4 (2)
(g) log1/3 (27)
(d) log 2 (16)
(h) log 5 (25)
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Section 5.3 & 5.4
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Author in Action: Determine the Domain of a Logarithmic Function (10:01)
Let’s recall the domain and range
of an exponential function:
Range
Domain
Domain
f ( x) = a x
All Real
Numbers
Since a logarithmic function is the inverse of an
exponential function, fill in the domain and range
below based on what we learned in Section 5.2.
Range
f ( x) = log a x
All Real
Numbers
greater
than 0
Domain and Range of the Logarithmic Function y = log a ( x) (defining equation x = a y )
Domain:__________________
Range:__________________
Example 4*: Determine the Domain of a Logarithmic Function
Find the domain of each logarithmic function.
x+3
(a) f ( x ) = log 3 ( x − 2 )
(b) F ( x ) = log 2
x −1
(c) h ( x ) = log 2 x − 1
(d) g ( x ) = log 1 x 2
2
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Exponential and Logarithmic Functions
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Author in Action: Graph Logarithmic Functions (13:44)
Post 5
1.
2.
3.
4.
5.
6.
Properties of the Logarithmic Function f ( x) = log a ( x)
The domain _______________; The range is _______________.
The x-intercept is _______________. There is _______________ y-intercept.
The y-axis ( x = 0 ) is a ____________________ asymptote of the graph.
A logarithmic function is decreasing if __________ and increasing if __________.
The graph of f contains the points ___________________________.
The graph is _______________________________, with no _________________.
Know natural and common logarithms
Fact
Natural Logarithm: ln ( x ) means log e ( x ) . It is derived from the Latin phrase, logarithmus naturalis.
In other words, y = ln( x) if and only if x = e y .
Copyright © 2016 Pearson Education, Inc.
Section 5.3 & 5.4
Example 5*: Graph Logarithmic Functions
(a)* Graph f ( x) = 3ln( x − 1) .
(b)* State the domain of f ( x ) .
(c)* From the graph, determine the range and vertical asymptote of f.
End of Post 5
(d) Find f −1 , the inverse of f.
(e) Use f −1 to confirm the range of f found in part (c). From the domain of f, find the range of f −1 .
(f) Graph f −1 on the same set of axis as f.
Know natural and common logarithms
Fact
Common Logarithm: log ( x ) means log10 ( x ) . In other words, y = log( x) if and only if x = 10 y .
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Exponential and Logarithmic Functions
Example 6: Graph a Logarithmic Functions
(a) Graph f ( x ) = −2log ( x + 2 ) .
(b) State the domain of f ( x ) .
(c) From the graph, determine the range and vertical asymptote of f.
(d) Find f −1 , the inverse of f.
(e) Use f −1 to confirm the range of f found in part (c). From the domain of f, find the range of f −1
(f) Graph f −1 on the same set of axis as f.
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Section 5.3 & 5.4
We will return to this next week.
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Author in Action: Solve Logarithmic Equations (6:43)
Solving Basic Logarithmic Equations
When solving simple logarithmic equations (they will get more complicated in Section 5.6) follow
these steps:
1. Isolate the logarithm if possible.
2. Change the logarithm to exponential form and use the strategies learned in Section 5.3 to solve
for the unknown variable.
Example 7*: Solve Logarithmic Equations
Solve the following logarithmic equations
(a)* log2 ( 2x +1) = 3
(b)* log x 343 = 3
(c) 6 − log(𝑥) = 3
(d) ln ( x ) = 2
(f) log 6 36 = 5 x + 3
(e) 7 log 6 (4 x) + 5 = −2
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Exponential and Logarithmic Functions
Steps for solving exponential equations of base e or base 10
1. Isolate the exponential part
2. Change the exponent into a logarithm.
3. Use either the “log” key (if log base 10) or the “ln” (if log base e) key to evaluate the variable.
Example 8*: Using Logarithms to Solve Exponential Equations
Solve each exponential equation.
(a) e x = 7
(b)* 2e3 x = 6
(c) e5 x −1 = 9
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