EXPONENTS & LOGARITHMS LEARNING PACKET (3)...2 1 ( ). Sketch the asymptote as a broken line and...
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PRE-IB HONORS ALGEBRA 2 HONORS ALGEBRA 2 EXPONENTS & LOGARITHMS LEARNING PACKET (3)
Transcript of EXPONENTS & LOGARITHMS LEARNING PACKET (3)...2 1 ( ). Sketch the asymptote as a broken line and...
PRE-IB HONORS ALGEBRA 2
HONORS ALGEBRA 2
EXPONENTS & LOGARITHMS
LEARNING PACKET (3)
HONORS ALGEBRA 2
GRAPHS OF EXPONENTIAL FUNCTIONS
Calculators Permitted
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**********
Learning Target: I will be able to sketch the graph of exponential functions to
include:
Describe the transformations from the parent function
Determine and sketch the horizontal asymptote
Give the domain and range
Describe the end behavior
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**********
A. PARENT EXPONENTIAL FUNCTIONS
1. The exponential parent function is ____________.
2. It is an exponential function because
_________________________________________
3. The __________________________________________ is a significant
characteristic of
exponential graphs. The horizontal asymptote for the parent exponential function is
________
4. Sketch y = 2x
Domain
____________
Range
____________
End behavior
__________________
__________________
5. This is an ___________________________________________curve.
6. We will also study _________________________________________
curves.
Sketch the graph of
x
xh
2
1)( . Sketch the asymptote as a broken line and label
with its equation.
Give the domain, range, and end behavior.
Domain
____________
Range
____________
End behavior
__________________
__________________
7. What is the difference in the equations of an exponential growth and an exponential
decay curve?
____________________________________________________________
______
____________________________________________________________
______
(over)
B. TRANSFORMATIONS OF THE PARENT EXPONENTIAL FUNCTION
1. The general form of an exponential function is
_____________________________ where:
a) h is the ___________________________________
b) k is the ___________________________________
c) y = k is the ________________________________
Graph the function on the axes provided. Sketch the asymptote as a broken
line and label with its equation. Give the domain, range, and end behavior.
2. f(x) = 3x – 2
Domain
____________
Range
____________
End behavior
__________________
__________________
3. 13
12
x
y
Domain
____________
Range
____________
End behavior
__________________
__________________
Student ___________________________
ASSIGNMENT 8.1
GRAPH EXPONENTIAL FUNCTIONS
Calculators permitted
1. Given: f(x) = (2)x – 4 – 3
a) Give the equation of the asymptote
____________
b) Find the y-intercept algebraically. Calculators not permitted. Show all work for
common
denominators.
____________
State the transformation(s) from the parent function, and then sketch the
graph. Sketch asymptotes as broken lines and label with their equations.
Give the domain, range, and end behavior.
2. f(x) = 5(x – 3)
a) ________________
3. f(x) = ½ (x + 2) + 4
a) _________________
b) _________________
4. Find the y-intercept of the graph of the function 35)( 43 xxf . Calculators
not permitted;
show all work for common denominators
b) Domain: ________________
c) Range: ________________
d) End behavior: _____________
________________
c) Domain: ________________
d) Range: ________________
e) End behavior: _____________
________________
____
____
____
5. If the simplified radical expression is aa
b 2
3 2 , which is a possible original
expression? ______
A 33
2
a
b B
3
2
9
2
a
b C
5
2
9
2
a
b D
5
2
18
2
a
b
6. If 3
3)(
x
xxf and 3)( xg , find f(x) – g(x)
____________
7. If 210)( 2 xxxf and xxg 2)( , find f(g(-5))
____________
HONORS ALGEBRA 2
GRAPH LOGARITMIC FUNCTIONS
LECTURE
Calculators Permitted
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**********
Learning Targets - I will be able to:
Graph logarithmic functions through transformations of the parent
function
Rewrite from exponential ↔ logarithmic form
Evaluate log expressions
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***********
A. RELATIONSHIP BETWEEN EXPONENTS & LOGARITHMS
1. Exponentials and logarithmic functions are ________________________
This means:
a) Their graphs are reflections of each other about the line ________________
b) If the point (x, y) is on the graph of f(x), then the point ____________ is on the
graph of f -1 (x).
B. PARENT LOGARITHMIC FUNCTIONS
1. The logarithmic parent function is _______________ which means
___________________
2. The __________________________________________ is a significant
characteristic of
logarithmic graphs. The vertical asymptote for the log parent function is
____________
3. Sketch y = log x
Domain
____________
Range
____________
End behavior
__________________
__________________
B. TRANSFORMATIONS OF THE PARENT LOGARITHMIC FUNCTION
1. The general form of a logarithmic function is
_____________________________ where:
a) h is the ___________________________________
b) k is the ___________________________________
c) x = h is the ________________________________
(over)
Graph the function on the axes provided. Sketch the asymptote as a broken
line and label with its equation. Give the domain, range, and end behavior.
2. f(x) = 2log 3 x
Domain
____________
Range
____________
End behavior
__________________
__________________
3. g(x) = 1)2(log4 x
Domain
____________
Range
____________
End behavior
__________________
__________________
C. REWRITE EXPONENTIAL ↔ LOGARITHMIC FORM
1. Exponents and logarithms are ________________
2. The definition of a log is generalized by this relationship:
_____________________________________________
Examples:
Exponential Form Logarithmic Form
3. 2x = 3 3.
4. 4x = 5 4.
5. 16 = 3x 5.
6. 6. 10log 5 x
7. 7. x3log2.7
8. 12 = 2x – 3 8.
D. EVALUATE LOG EXPRESSIONS
Evaluate to two decimal places.
1. log 35 ____________ 2. ln 15
____________
3. Change of Base property
a) Use when
___________________________________________________
________
b) __________________________
4. 4
log 7 ____________ 5. 6
log 2
____________
Student _______________________
ASSIGNMENT 8.2
GRAPH LOG FUNCTIONS
Calculators permitted
1. Give the equation of the asymptote: f(x) = 1)4log( x
____________
Matching. Match the sketch with the appropriate function. Two answers
will not be used.
2. ______ 3. ______ A. cubic
B. absolute
value
--------------------
C. quadratic
4. ______ 5. ______
D.
exponential growth
E.
exponential decay
6. ______ 7. ______ F. square
root
G.
logarithmic
--------------------- H. cube root
Use the sketch to answer questions 8 & 9.
8. Give the solution(s) to the quadratic equation
________
9. Give the solution(s) to the system
____________
Rewrite from exponential ↔ logarithmic form.
10. 5.4log 2 x ____________ 11. 3x = 6.2
____________
12. 6 = 2x ____________ 13. 12 = log5 x
____________
(over)
State the transformation(s) from the parent function, and then sketch the
graph. Sketch asymptotes as broken lines and label with their equations.
Give the domain, range, and end behavior.
14. h(x) = log (x + 1) + 2 15. f(x) = log5 (x) – 4
Transformation(s): Transformation(s):
a) __________________________ a)
___________________________
Use the graphic calculator to evaluate the expression exactly or to two
decimal places. Show change of base if required.
b) Domain: ________________ b) Domain: ______________
c) Range: ________________ c) Range: _______________
d) End behavior: d) End behavior:
_______________________ ____________________
_______________________ ____________________
16. log (2.5) ____________ 17. 34.1
____________
18. 30
2110
____________ 19. 3 ln 6 + 5
____________ (write in standard form)
20. log7 22 ____________ 21. 3log 5 +7
____________
change of base: change of base:
HONORS ALGEBRA 2
GRAPHS OF NATURAL BASE
EXPONENTIAL & LOGARITHMIC FUNCTIONS
A. REVIEW
Use the function 12)(2 x
xf to answer questions 1 – 3.
1. This is an example of an ________________________________ curve.
2. The base is ____________
3. The equation of the _________________ asymptote is ____________. horizontal/vertical
Use the function )4log()( xxh to answer questions 4 – 6.
4. This is an example of an ________________________________ curve.
5. The base is ____________
6. The equation of the _________________ asymptote is ____________. horizontal/vertical
Use the function xxg2
13)( to answer questions 7 – 9.
7. This is an example of an ________________________________ curve.
8. The base is ____________
9. The equation of the _________________ asymptote is ____________. horizontal/vertical
B. NATURAL BASE FUNCTIONS
1. The natural base is ________ ≈ ________
2. The natural base exponential parent function is ___________________
3. The general form of a natural base exponential function is ________________
Again, h and k are ________________________ of the parent function and
the equation of the ________________________ asymptote is ________.
4. a) A common log has ______________ denoted ________ or ________
b) A natural log has _____________________denoted __________
5. The natural base logarithmic parent function is ___________________
6. The general form of a natural log is ________________________
Again, h and k are ___________________ of the parent function and the
equation of the ____________________ asymptote is ____________.
C. GRAPH
Graph the following functions on the grids provided. Sketch the asymptote as a broken line and label with its equation. Give the domain, range and end behavior.
1. h(x) = e (x + 2) 2. f(x) = e (x – 3) – 4
3. h(x) = ln (x + 1) + 2 4. f(x) = ln (x – 3)
Domain: ________________ Domain: _________________
Range: ________________ Range: _________________
End behavior: End behavior
________________ _________________
________________ _________________
Domain: ________________ Domain: _________________
Range: ________________ Range: _________________
End behavior: End behavior
________________ _________________
________________ _________________
Student ___________________
HONORS ALGEBRA 2
ASSIGNMENT 8.3
GRAPH EXPONENTIAL & LOGARITHMIC NATURAL BASE FUNCTIONS
1. If the domain of g(x) = 3(x – 1) – 4 is {0, 1, 2}, what is the range? __________________
2. Given g(x) = 3x + 4. Will the graph of g(x) have a zero? Why or why not?
___________________________________________________________________
___________________________________________________________________
3. Given f(x) = 3)1log( x
a) Explain how f has been transformed from the function g(x) = 5)3log( x
_________________________________________________________________
b) The point (2, -3) is on the graph of f . What point is on the graph of f -1 ? ____________
4. Give the mathematical term for the relationship between exponential and logarithmic functions
and explain its meaning.
___________________________________________________________________
5. Give the first and last name of the mathematician for whom “e” is named. ________________
6. e ≈ ____________
7. Explain the difference between common and natural logs.
___________________________________________________________________
8. Give an exponential function with asymptote y = 2. __________________
9. Give a natural base logarithmic function with asymptote x = -4. __________________
10. Given the graph of g(x) below. Sketch the graph of g -1(x) on the axes beside it.
(1, 5)
(0, 3)
y = 2
(over)
Give the transformation from the parent function, and then sketch the graph. Sketch the asymptote as a broken line and label with its equation. Give the domain, range, and end behavior.
11. f(x) = e (x + 3)
____________________
12. f(x) = e x – 5 + 2
____________________
____________________
13. h(x) = ln (x – 2) + 1 14. f(x) = ln (x + 1) – 2
Domain: ________________ Domain: _________________
Range: ________________ Range: _________________
End behavior: End behavior:
________________ _________________
________________ _________________
Domain: ________________
Range: ________________
End behavior: ____________
____________
Domain: ________________
Range: ________________
End behavior: ____________
____________
HONORS ALGEBRA 2
SOLVE EXPONENTIAL & LOGARITHMIC EQUATIONS
A. Review: exponential & logarithmic functions
Given: f(x) = 2x
1. This is an exponential function because ________________________________________
2. The equation 2x = 8 is easy to solve by inspection: x = ________. But, what if you were asked to
solve the equation 2x = 12.6?
3. To find the exact value of an exponent, logarithms must be used.
Remember: LOGARITHMS ARE ________________.
4. Exponents and logarithms are _____________________________________.
5. The definition of logarithms is generalized by this relationship:
________________________________________________
6. Rewrite exponential ↔ logarithmic form
a) log3 9 = 2 ____________ b) ln 6 ≈ 1.79 ____________
c) 42 = 16 ____________ d) e4 ≈ 54.6 ____________
B SOLVE EXPONENTIAL EQUATIONS: ____________________________________
1. 10x + 5 = 60 ____________ 2. 4e2x = 5 ____________
3. 2(4)3x + 1 = 11 ____________ 4. 12 + 3(3).1x = 18 ____________
C. SOLVE LOGARITHMIC EQUATIONS: _________________________________
1. 2 log (3x) = 4 ____________ 2. -6 + log4 (2x) = 1 ____________
3. -5 + 3 ln x = 6 ____________
D. SUMMARY
________________________________________
________________________________________
________________________________________
E. Exponential and logarithmic applications
________________ ________________ ________________
________________ ________________ ________________
where A is ________________________ P is ______________________
r is ___________________ n is ___________________________
t is _______________________
trPA )1(
nt
n
rPA
1
rtPeA
1. Interest. A sum of $9 000 is invested at an annual rate of 8.5%. Find the balance to the
nearest dollar after 10 years if the account is compounded:
a) monthly ____________
b) continuously ____________
2. Exponential growth. If The price of oil change for your car is presently $23.95, estimate the
price 10 years from now if the annual rate of inflation averages 4% over the next 10 years.
____________
3. Exponential Growth. The approximate number of fruit flies in an experimental population
after t hours is given by
Q(t) = 20e 0.03t
a) How many fruit flies will there be after 60 hours? ____________
b) Determine how long it will take to reach 250 fruit flies. ____________
4. Inverse functions
a) If f(x) = 3(x – 5) + 2, find f -1(x). ____________
b) If g(x) = log4(x + 5) – 7, find g -1(x). ____________
Student _______________________
HONORS ALGEBRA 2
ASSIGNMENT 8.4
Solve the exponential equation. Round non-integral answers to two decimal places.
1. 2x = 15 ____________ 2. 4x – 5 = 3 ____________
3. 30.1x – 4 = 8 ____________ 4. 51)4( 2
4
1 x ____________
5. -5e–x + 9 = 6 ____________ 6. 43
14
3
2 xe ____________
(eliminate fractions first)
Solve the logarithmic equation. Round non-integral answers to two decimal places.
7. 4 log3 x = 28 ____________ 8. 3)16(log 62
1 x ____________
(over)
trPA )1(
nt
n
rPA
1
rtPeA
9. 2 ln(-x) + 7 = 14 ____________ 10. -5 + 2.3 ln (3x) = 5 ____________
You invested $5 300 in a savings account at an annual rate of 7½ %. Find the balance
to the nearest cent after 5 years if the interest is compounded:
11. daily ____________
12. continuously ____________
13. If the account is compounded continuously, determine algebraically how long it will take to
reach a balance of $10 000.
____________
Suppose you purchase a new car for $20 000 in 2000. The value of the car decreases by
16% each year. The following exponential decay model represents this situation.
A = 20 000(1 – 0.16)t, t = 0 represents 2000.
where A represents the value of the car and t represents the time in years.
14. Find the value of the car in 2004 algebraically. ____________
15. Determine algebraically the year in which the car will be worth approximately $2 500.
____________
In a bacterial culture, the following model can be used to represent the number of
bacteria:
B = Pe .45t
where B represents the number of bacteria, P represents the initial number of
bacteria, and t represents the time in hours.
16. If the bacterial count was 12 000 at 8:00 AM, determine algebraically what the count would
be at 1:30 PM.
____________
17. Determine the time of day in hours and minutes when there will be approximately 100 000
bacteria.
____________
For problems 8 & 9, find inverse functions.
18. If f(x) = 3(2)(x + 3) – 4, find f -1(x). ___________________
19. If g(x) = log3(x – 7 ) + 5, find g -1(x). __________________
Student ________________
HONORS ALGEBRA 2
QUIZ 8
PART A. Calculators Not Permitted
1. Which function represents the graph of an exponential decay? ______
A 2
213)(
xxf B 4)log()( xxg C h(x) = ln (x – 4) D k(x) = ex – 2 + 4
2. Which correctly shows the change of base formula for )2(log kd ? ______
A 2ln
ln
d
k B 2
ln
ln
k
d C
d
k
ln
)2ln( D
)2ln(
ln
k
d
For problems 3 & 4, sketch the graph the function on the axes provided. Give the
domain, range, and end behavior.
3. f(x) = log(x – 3) + 3
Domain ____________
Range ____________
End behavior __________________
__________________
4. g(x) = e(x + 2)
Domain ____________
Range ____________
End behavior __________________
__________________
5. For f(x) = 5log x , which describes the behavior of x as y approaches -∞ ? ______
A x approaches -∞ B x approaches 0
C x approaches 5 D x approaches ∞
6. Which describes the end behavior of 4)2(3)( 1 xxg as x approaches -∞ ? ______
A g(x) approaches -∞ B g(x) approaches ∞
C g(x) approaches 1 D g(x) approaches 4
(over)
7. Given the graph of f(x) below. Sketch the graph of f -1(x) on the axes beside it. Label the points
with their coordinates and the asymptote with its equation.
(0, 5)
(-2, 0)
x = -3
8. Give the transformations from the pre-image f '= 4x – 1
+ 2 to the image f = 4x – 6
– 7.
_______________________________________________________________________________________
For problems 9 & 10, rewrite from exponential ↔ logarithmic form.
9. 4 = e x – 5 ____________ 10. 12 = log (x – 2) ____________
Use the following functions to answer questions 11 & 12.
A 33)( xxf B 33)( xxg C 3)(21
xxh D 3ln)( xxj
E 3)3()( 2 xxk F 3
1)(
xxp G 3)( xesr H )3log()( xst
11. Which functions have asymptote x = 3 ? __________________
12. Which functions have a limited domain: i.e. not “all real numbers”? __________________
13. Given f(x) = 4(2)x + 4, find f -1 (x). __________________
14. Given g(x) = ln (x + 2) – 5, find g -1(x) __________________
PART B. Calculators Permitted.
Solve for x. Round all non-integer answers to two decimal places.
15. 20ln5 2 x ____________ 16. 2 + log 2x = 5.4 ____________
17. 3(x + 2) – 2 = 12 ____________ 18. ½ e .4x = 10 ____________
19. You deposit $10 000 in an account that pays 6% interest.
a) Find the amount in the account to the nearest dollar after 12 years if the interest is
compounded quarterly.
____________
b) Find the amount in the account to the nearest dollar after 12 years if the interest is compounded continuously.
____________
c) Determine algebraically the number of years until there is $30 000 in the account if the account is compounded continuously.
____________
20. The following model can be used to determine the number of bacteria in a culture:
B = 325e .4t
where B represents the number of bacteria, P represents the initial number of bacteria, and t
represents the time in hours.
a) Find the number of bacteria after 6 ½ hours. ____________
b) Determine algebraically the number of hours needed to have a culture containing 10 000 bacteria.
____________
21. You purchased a house in 2005 for $125 000 that is predicted to increase in value 8½ % per year according to the model
A = 125 000(1.085)t
where A represents the amount after time, and t represents the time in years.
a) Determine the value of your house to the nearest dollar after 10 years.____________
b) Determine algebraically how many years must pass before your house is worth $400 000.
____________
