Download Exponential Functions and Their Graphs

Exponential Functions and Their Graphs
Definition of Exponential Function
f ( x)  a
x
where a > 0, a ≠ 1, and x is any
real number
Parent function for
transformations:
f(x) = 𝑏 ∙ 𝑎
𝑥 −ℎ
+𝑘
The graph of f moves
to the left if h < 0 or
to the right if h > 0.
The graph of f moves
upward if k > 0 or
downward if k < 0.
The graph of f is
stretched if b > 1 and
shrunk if 0 < b < 1.
The graph of f is
reflected in the x-axis
if b is negative.
Properties of Exponential Functions
The parent graph of the exponential function
f(x) = ax has the following properties:
1.
2.
3.
4.
5.
The function f is increasing for a > 1 and
decreasing for 0 < a < 1.
The y-intercept of the graph of f is (0, 1).
The graph has the x-axis as a horizontal
asymptote.
The domain of f is (–∞, ∞), and the range of f is
(0, ∞).
The function f is one-to-one.
Slide 4-3
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f(x) = ax
for a > 1
y
f(x) = ax
for 0 < a <1
11
1
x
Transformations of Graphs of Exponential Functions
Describe the transformations of each function from the
parent graph of 𝑓 𝑥 = 𝑎 𝑥 .
1.
f ( x)  3  7
x
2.
𝑓 𝑥 = 2𝑥−3 + 5
Sketch the graphs of #1 and #2, be sure to include
accurate points and any asymptotes.
3.
4.
Transformations of Graphs of Exponential Functions
Describe the transformations of each function from the
parent graph of 𝑓 𝑥 = 𝑎 𝑥 .
5. f(x) =
1 𝑥−3
4
Sketch the graph.
7. f(x) =
1 𝑥+1
4
−2
6. f(x) = −
1 𝑥+4
4
−1
One-to-One Property of
Exponential Functions:
For a > 0 and a ≠ 1,
if a  a , then x1  x2 .
x1
x2
Slide 4-7
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Solve each equation:
8. 2𝑥 = 27
10. 2 =
𝑥
1
2
9. 2𝑥 =64
11. 2𝑥+3 = 24𝑥−9
Compound Interest
 r
A  P 1  
 n
nt
n: number of times compounded in a
year
A  Pe
rt
Used when interest
is continuously
compounded
r: interest rate as a decimal
t: time in years
r: interest rate as a
decimal
P: principal (initial investment)
t: time in years
A: amount in the account after time(t)
P: principal (initial
investment)
12. If you invest $1000 for 25 years and the interest rate
is 6%, what is the value of your investment if your
interest is compounded quarterly?
13. If you invest $5000 for 18 years and the interest rate
is 4.5%, what is the value of your investment if your
interest is compounded continuously?
Section 4.2
Logarithmic Functions
and Their Applications
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Since exponential functions are one-to-one functions,
they
are invertible. The inverses of the exponential functions
are
called logarithmic functions.


If f(x) = ax, then instead of f –1(x), we write loga(x) for
the inverse of the base-a exponential function.
We read loga(x) as “log base a of x,” and we call the
expression loga(x) a logarithm.
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412
Definition of Logarithmic Function with Base a
Definition: Logarithmic Function
For a > 0 and a ≠ 1, the logarithmic function with
base
a is denoted as f(x) = loga(x), where
y = loga(x)
Logarithmic Form
y  log a x
if and only if
ay = x.
Exponential Form
a x
EQUIVALENT!
y




Note that loga(1) = 0 for any base a, because a0 = 1
for any base a.
There are two bases that are used more frequently
than others; they are 10 and e.
The notation log10(x) is abbreviated log(x) and loge(x)
is abbreviated ln(x). These are called the common
logarithmic function and the natural logarithmic
function, respectively.
There are no logarithms of negative numbers or zero.
Slide 4-14
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Any function of the form
k
g(x) = b · loga(x – h) +
is a
member of the logarithmic family of functions.




The graph of f moves to the left if h < 0 or to the right
if h > 0.
The graph of f moves upward if k > 0 or downward if k
< 0.
The graph of f is stretched if b > 1 and shrunk if 0 < b
< 1.
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The graph of f is reflected in the x-axis if b is negative.
Properties of Logarithmic Functions
The logarithmic function f(x) = loga(x) has the following
properties:
1.
The function f is increasing for a > 1 and decreasing for
0 < a < 1.
2.
The x-intercept of the graph of f is (1, 0).
3.
The graph has the y-axis as a vertical asymptote.
4.
The domain of f is (0, ∞), and the range of f is (– ∞, ∞).
5.
The function f is one-to-one.
6.
The functions f(x) = loga(x) and f(x) = ax are inverse
functions.
for a > 1
Slide 4-17
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y = loga x
for 0 < a < 1
(1, 0)
1
Slide 4-18
Describe the transformations of each function from the
parent graph. Then sketch the graph.
One-to-One Property of
Logarithms
For a > 0 and a ≠ 1,
if
loga(x1) = loga(x2),
then
x1 = x2 .
Section 4.3 & 4.4
Properties of Logarithms
&
Solving Logarithmic
Equations
Use the Change of Base Formula to evaluate.
Round your answers to the nearest thousandth.
26.
log 3 16 
27.
log 50 5 
Find the exact value of each expression without
using a calculator
30.
28.
29.
5
log7 7
ln e  ln e
12
5
0.2
log3 81
Expand each logarithmic expression
31.
xy
log 4
z
4
32.
3
ln 5 x
2
Condense each logarithmic expression
33.
2ln8  5ln( z  4)
34.
2[3log x  log 6  log y]
Strategies for Solving Exponential
and Logarithmic Equations
• Try to use the one-to-one properties (match them up)
• Condense to single logarithms when possible
• Switch from exponential form to logarithmic form
• Switch from logarithmic form to exponential form
Exponential Equations-solve for x, check answers
35.
1.
4(3 )  64
x
36.
2.
e
 x2
5 x 6
e
Exponential Equations-solve for x, check answers
37.
3.
5  3e  2
x
38.
4.
62
x 5
  4  11
Logarithmic Equations-solve for x, check answers
39.
40.
log 5 = 2 − log(𝑥)
ln 𝑥 + ln 𝑥 + 2 = ln(8)
Logarithmic Equations-solve for x, check answers
3. log3 (5x  13)  log3 6  log3 3x
41.
6. log x  log( x  9)  1
42.
43.
𝑙𝑜𝑔4 𝑥 − 𝑙𝑜𝑔4 𝑥 + 2 = 2
Compound Interest Formula
If a principal P is invested for t years at an annual rate r
compounded n times per year, then the amount A, or
nt
r

ending balance, is given by A  P1   .
 n
Continuous Compounding Formula
If a principal P is invested for t years at an annual rate r
compounded continuously, then the amount A, or
ending balance, is given by A = Pert.
44.
Melinda invests her $80,000 lottery
winnings at a 9% annual interest
rate compounded monthly. How
long much she invest the money to
have $480,000 in her account.
Round to the nearest year.
45.
At what interest rate would a
deposit of $30,000 grow to
$2,540,689 in 40 years with
continuous compounding?
Round to the nearest tenth of a
percent.
Systems of Equations
and Inequalities
Section 5.1
Systems of Linear
Equations in Two
Variables
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Education, Inc.


Any collection of two or more equations is called a
system of equations.
The solution set of a system of two linear equations
in two variables is the set of all ordered pairs that
satisfy both equations of the system.
◦ The graph of an equation shows all ordered pairs
that satisfy the equation, so we can solve some
systems by graphing the equations and observing
which points (if any) satisfy all of the equations.
THESE SOLUTION WOULD BE WHERE THE GRAPHS
INTERSECT.
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◦ A system of equations that has at least one solution
is consistent. Two types of consistent systems are:
 Independent - with exactly one solution
Two Intersecting Lines
 Dependent - with infinitely many solutions
Same Lines
◦ A system with no solutions is inconsistent.
Parallel Lines
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◦ In the substitution method, we eliminate a
variable from one equation by substituting an
expression for that variable from the other
equation.
◦ In the addition method, (also called the
elimination method) we eliminate a variable by
adding the two equations.
 It might be necessary to multiply each equation by an
appropriate number so that a variable will be
eliminated by this addition.
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#46.
𝑦 = 2𝑥 + 1
3𝑥 − 4𝑦 = 1
#49. −2𝑥 + 5𝑦 = 14
7𝑥 + 6𝑦 = −2
#47.
𝑦 − 3𝑥 = 5
3𝑥 − 3 = 𝑦 − 2
#48.
2𝑥 + 𝑦 = 9
4𝑥 + 2𝑦 = 10


#50. Amy has a higher income than Vince,
and their total income is $82,000. If their
salaries differ by $16,000, then what is the
income of each?
#51. The Springfield zoo has different
admission prices for adults and children. When
Mr. and Mrs. Weaver went with their five
children, then bill was $33. If Mrs. Wong and
her three children got in for $18.50, then what
were the individual prices for adult and
children’s tickets?
Matrices and
Determinants
Section 6.1
Solving Linear
Systems Using
Matrices
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


A matrix is a rectangular array of real numbers.
The rows of a matrix run horizontally, and the columns
run vertically.
A matrix with m rows and n columns has size m  n (read
“m by n”).
◦ The number of rows is always given first.
Columns
Rows
This is a 2 x 3 matrix.
Both equations in the system need to be in standard form: Ax + By = C
2x – 3y = 6
y= x +4
Already in standard form.
Put into standard form.
2x – 3y = 6
-x + y = 4
2x – 3y = 6
-x + y = 4
Put the values of
A, B and C into
the matrix.
2 −3 6
−1 1 4
*See handout for specific instructions on
using the calculator to now solve the system
of equations. It is on the CAS website
Check solutions to #46 – 49
using the matrix capabilities of
your calculator.