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Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is
1. Where possible, evaluate logarithmic expressions.
1
2
(log
3X +
(+ log 3y) – 5 log 3(x+5)
Ź (10g 3*+ log 3y) – 5log 3(x+5) = [
Find the difference quotient off; that is, find
f(x+h)-f(x)
h
h70, for the following function. Be sure to simplify.
1
f(x) = x2 – 9x + 2
f(x+h)-f(x)
h
(Simplify your answer.)
ob
Use the vertex and intercepts to sketch the graph of the
quadratic function. Give the equation for the parabola’s axis of
symmetry. Use the parabola to identify the function’s domain and
range.
N
6-
5-
4-
f(x) = x2 + 5x+6
Use the graphing tool to graph the equation. Use the vertex and
one of the intercepts to graph the equation.
Click to
enlarge
graph
3-
2-
1-
х
-6
-4
-2
2
4
6
The axis of symmetry is
(Type an equation.)
Identify the function’s domain.
The domain is
(Type the answer in interval notation.)
1 2
Identify the function’s range.
The range is
(Type the answer in interval notation.)
Begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function.
g(x) = – |*-2 +6
What transformations are needed in order to obtain the graph of g(x) from the graph of f(x)? Select all that apply.
A. Vertical stretch/shrink
B. Horizontal translation
C. Reflection about the y-axis
D. Horizontal stretch/shrink
E. Reflection about the x-axis
F. Vertical translation
Choose the correct graph of g(x) below.
OA.
B.
C.
OD
10-
-10
10
-10
10
10
10
10
-108
-10
– 10
-105
Ау
104
х
-10
10
-10
104
X
-10
10
-10
10-
X
-10
10
워
-10-
Ау
10-
х
-10
10
-10-
Verify the identity
sin
31
2
–
cos e
Write the left side of the identity using a sum or difference formula for sine or cosine.
(Do not simplify.)
The expression from the previous step then simplifies to – cos 0 using what?
O A. Even-Odd Identity
B. Reciprocal Identity
OC. Quotient Identity
OD. Pythagorean Identity
O E. No identity is required. Evaluate the previous expression wherever possible and simplify.
X + 1 if x2
The domain of the piecewise function is (-00,00).
a. Graph the function.
b. Use your graph to determine the function’s range.
f(x) =
X-1 if x22
a. Choose the correct graph below.
OA.
ОВ.
OC.
OD.
10-
10-
I
10-
х
X
х
16
-10
10
-10
LLA
S
10
10
-10
10
3
2
2
-10-
-10
b. The range of f(x) is
(Type your answer in interval notation.)
10-
х
-10
10
-10-
10-
х
-10
10
-10-
104
х
-10
10
-10
10-
X
-10
10
– 10
The graph of a quadratic function is given. Write the function’s equation, selecting from the following options.
g(x) = x2 – 12x + 36 j(x)= – x2 – 36 f(x) = x2 + 12x + 36 h(x)= x2 – 36
Drag each function given above into the area below the appropriate graph, depending on which function is represented by which
graph.
1.
2.
80
AY
89-
40H
40
-80 -40
-40
40 80
-80-40
-40-
40 80
3
3
-80
-801
3.
4.
807
AY
80-
40-
40-
-80 -40
40 80
-80-40
-401
40 80
-40-
LY
-80-
800
Ay
80-
60
40-
20-
X
-80
-60
-40
-20
20
40
60
80
-40-
-60-
-80
80
60-
40-
201
X
에
-80
-60
-40
-20
20
40
60
80
-20
-40
-60
-80-
89-
6CH
401
20-
X
-80
-60
-40
-20
20
40
60
80
-20
-40-
-60
-80-
80-
60-
404
20-
X
-80
-60
-40
-20
20
40
60
80
-20-
어
B01
The annual yield per walnut tree is fairly constant at 140 pounds per tree when the number of trees per acre is 30 or fewer. For each
additional tree over 30, the annual yield per tree for all trees on the acre decreases by 2 pounds due to overcrowding. How many
walnut trees should be planted per acre to maximize the annual yield for the acre? What is the maximum number of pounds of
walnuts per acre?
trees per acre to maximize annual yield for the acre.
Plant
The maximum annual yield is pounds of walnuts per acre.
Find the domain of the function.
4
f(x) =
x2 + 7x + 6
What is the domain off?
(Type your answer in interval notation.)
Give the slope and y-intercept of the line whose equation is
given. Then graph the linear function.
10-
ob
3
f(x)=2x-6
6-
4-
Select the correct choice below and, if necessary, fill in the
answer box to complete your choice.
2-
х
-10
-8
-6
-4
-2
2
01-
-00
10
O A. The slope is
(Simplify your answer. Type an integer or a fraction.)
B. The slope is undefined.
-2
-4-
Select the correct choice below and, if necessary, fill in the
answer box to complete your choice.
6 8
-10
O A. The y-intercept is
(Simplify your answer. Type an integer or a fraction.)
OB. There is no y-intercept.
Use the graphing tool to graph the equation.
Click to
enlarge
graph
Begin by graphing the square root function, f(x) = Vx. Then, use transformations of this graph to graph the given function.
h(x) = -x-3
Choose the correct graph below.
OA.
B.
OC.
OD.
х
х
х
-10
10
-10
10
-10
10
-10
2
LY
-10
-10
10-
х
-10
10
-10
10-4
-10
10
10-
у
104
х
-10
10
-104
Ау
104
х
-10
10
-10
Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form.
1
Passing through (7. – 8) and perpendicular to the line whose equation is y=-3x+1
3x
Write an equation for the line in point-slope form.
(Simplify your answer. Use integers or fractions for any numbers in the equation.)
Write an equation for the line in slope-intercept form.
(Simplify your answer. Use integers or fractions for any numbers in the equation.)
Find the domain of the following logarithmic function.
g(x) = In (x-6)2
The domain is
(Type your answer in interval notation.)
Find the exact value of each of the remaining trigonometric functions of e.
40
sin O=
41
in Quadrant II
cos =
(Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
tan =
(Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
csc =
(Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
seco=
(Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
cot O=
(Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
Follow the seven step strategy to graph the following rational function.
2x
f(x)
x² – 4
To graph the function, first determine the symmetry of the graph of f. Choose the correct answer below.
O y-axis symmetry
O neither y-axis symmetry nor origin symmetry
o origin symmetry
What is the y-intercept? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
O A. The y-intercept is
(Type an integer or a simplified fraction.)
B. There is no y-intercept.
What is/are the x-intercept(s)? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
O A. The x-intercept(s) is
(Type an integer or a simplified fraction. Use a comma to separate answers as needed.)
B. There is no x-intercept.
Follow the seven step strategy to graph the following rational function.
2x
f(x)
x² – 4
Find the vertical asymptote(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
O A. The equation(s) of the vertical asymptote(s) is/are
(Type an equation. Use a comma to separate answers as needed.)
OB. There is no vertical asymptote.
Find the horizontal asymptote(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
O A. The equation(s) of the horizontal asymptote(s) is/are
(Type an equation. Use a comma to separate answers as needed.)
OB. There is no horizontal asymptote.
Plot points between and beyond each x-intercept and vertical asymptote. Find the value of the function at the given value of x.
-6 -5 -1 1 6 8
2x
f(x) =
x² – 4
(Simplify your answer. Type an integer or a simplified fraction.)
х
Follow the seven step strategy to graph the following rational function.
f(x)
2x
x² – 4
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Use the information obtained in the previous steps to graph the function between and beyond the vertical asymptotes. Choose the
correct graph below.
ОА.
ОВ.
OC.
OD.
х
Q
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10
-10
10
-10
10
-10
10
LY
2
10
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10
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1
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1
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-10
10
1
-10
Determine the amplitude and period of the following function.
Then, graph the function.
104
8-
y = 8 COS TX
6-
The amplitude is
(Simplify your answer.)
4-
2-
х
Lo
-1.5
-1
-0.5
0.5
1.5
2
-2-
The period is
(Type an exact answer using a as needed. Use integers or
fractions for any numbers in the expression.)
Use the graphing tool to graph the function.
Click to
enlarge
graph
(For any answer boxes shown with the grapher, type an exact
answer. Type the word pi to insert the symbol a as needed.)
-4-
-6-
ob
-104
Solve the triangle.
c.
6
a=s
449
A A
B
с
O
A=
(Do not round until the final answer. Then round to the nearest degree as needed.)
b
(Do not round until the final answer. Then round to the nearest tenth as needed.)
(Do not round until the final answer. Then round to the nearest tenth as needed.)
Use a calculator with a y key or a key to solve the following.
The exponential function f(x) = 547(1.032)* models the population of a country, f(x), in millions, x years after 1975. Complete parts
(a)-(e).
a. Substitute 0 for x and, without using a calculator, find the country’s population in 1975.
The country’s population in 1975 was million
b. Substitute 22 for x and use your calculator to find the country’s population, to the nearest million in the year 1997 as modeled by
this function.
The country’s population in 1997 was million.
c. Find the country’s population, to the nearest million, in the year 2019 as predicted by this function.
The country’s population in 2019 will be million.
d. Find the country’s population, to the nearest million in the year 2041 as predicted by this function.
The country’s population in 2041 will be million.
e. What appears to be happening to the country’s population every 22 years?
O A. It appears that the population is growing by a factor of 3 every 22 years.
B. It appears that the population is growing by a factor of 2 every 22 years.
O C. There does not appear to be a pattern.
OD. It appears that the population is decreasing by a factor of
every 22 years
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Tags:
Properties of Logarithms
difference quotient
axis of symmetry
logarithmic expression
vertex and intercepts
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