Description
In This collection, we will go deep into math.
This collection will help all math and high school students.
Study Set Content:
Luttrell 2012
121
Name: ______________________
Date: _____
8h –Quadratic Formula
All quadratic equations are of the form
y
=
ax
2
+
bx
+
c
, where a, b, and c are real numbers.
When
y
= 0,
y
ou can solve for
x
by factoring or using the quadratic formula. Factoring doesn’t
alwa
y
s work, but the quadratic formula will!
Quadratic Formula:
x
b
b
ac
a
=
− ±
−
2
4
2
E
x
ample A:
x
2
+ 2
x
-3 = 0
Identif
y
parts: a = 1, b = 2, c = -3
Substitute into formula:
x
=
− ±
−
−
=
− ±
=
− ±
2
2
4 1
3
2 1
2
16
2
2
4
2
2
( )(
)
( )
= 1 or -3
Find the solutions (roots, zeros,
x
-intercepts):
1. 5
x
2
+2
x
- 3 = 0
2. 4
x
2
- 3
x
- 7 = 0
3. 4
x
2
+5
x
- 6 = 4
4. 3
x
2
+ 7
x
- 10 = 0
5.
x
2
- 5
x
+ 6 = 0
6. 3
x
2
+ 11
x
- 4 = 0
7. x
2
- 5x - 6 = 0
8. x
2
+ 10x + 21 = 0
9. x
2
- 2x + 2 = 2x
Luttrell 2012
122
Name: ______________________
Date: _____
8i – more Quadratic Formula
Solve for the given variable, leaving answer in simplest form.
1. 3
x
2
- 5
x
- 8 = 0
2. w
2
+ 5w - 6 = 0
3. z
2
+ 7z - 8 = 0
4. g
2
+ 5g - 5 = 0
5.
y
2
- 2
y
- 3 = 0
6. 4
x
2
+ 10
x
- 14 = 0
Solve for
x
:
7. 3a
2
+ 10a + 5 = 0
8. -2
x
2
+ 5
x
- 3 = 0
bonus: 3
x
2
+ a
x
- 4a
2
= 0
Luttrell 2012
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Name: ______________________
Date: _____
8j – Graphing Quadratics
Polynomials with a general form of
y
=
ax
2
+
bx
+
c
are called quadratic equations. When a
quadratic is graphed, the shape of the curve is referred to as a parabola. One of the easiest, but
time-consuming methods of graphing a parabola is to complete a t-chart. Then plot the points.
E
x
ample: Graph
y
= 2
x
2
- 3
x
+ 1.
Start by choosing common values for x or y:
In the end it should look like:
Substitute the values you picked:
When
x
= -1, then
y
= 2 + 3 + 1 = 6.
When
x
= 0, then
y
= 0 - 0 + 1 = 1.
When
x
= 1, then
y
= 2 - 3 + 1 = 0.
When
y
= 0, then 0 = 2
x
2
- 3
x
+ 1.
Then b
y
factoring, we can solve 0 = (2
x
-1)(
x
-1) for
x
= ½ or 1.
Place the corresponding values in the t-chart. Then plot on the
axis provided. Connect the dots. The shape of the curve should
look like a
u.
Fill in the t-charts. Then graph to the right. Please label!
1.
y
=
x
2
+ 4
x
- 5
x
-5
-3
-1
0
1
3
y
2.
y
= 2
x
2
+ 3
x
- 5
x
-3
-2
-1
0
1
2
y
3.
y
=
x
2
+ 5
x
+ 6
x
- 4
-3
-2
-1
0
1
y
4.
y
=
-x
2
+ 3
x
+ 4
x
-1
0
1
2
3
4
y
x
-1
0
½
1
2
y
6
1
0
0
3
x
-1
0
1
2
y
0
Luttrell 2012
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Name: ______________________
Date: _____
8k – Graphing Quadratics
Parabolas have certain defining characteristics. If we know those characteristics, then we can
use them to make graphing easier. One of the first characteristics to see is the Line of s
y
mmetr
y
.
If you ‘cut’ down that line,
y
ou would cut the parabola in halves. And each half would be a
reflection of the other. If your parabola was alread
y
graphed (see #4 from lesson 8j), then take
two ordered pairs with the same
y
-value. Find the midpoint between those ordered pairs. The
line of s
y
mmetr
y
will pass through this point so that it cuts the parabola in reflected halves. The
line of s
y
mmetr
y
will also pass through the verte
x
, which is lowest point or highest point on the
graph. If the parabola were a string necklace, the verte
x
would be where the single charm would
hang. Without the graph, some refer to the equation
y
=
ax
2
+
bx
+
c
and determine the verte
x
x
-
coordinate with the formula:
x
b
a
= −
2
. Then the
y
can evaluate for the
y
-coordinate. Another
quick calculation is done to find the
y
-intercept, where the parabola crosses the
y
-a
x
is. Evaluate
for
y
when
x
= 0.
Y
ou’ll notice that
y
-value is the same as
c
in
y
=
ax
2
+
bx
+
c
. Reflect the
y
-
intercept over the line of symmetry to get another point. With those three points,
y
ou can graph.
You can always check
y
our work by knowing the direction of the parabola. From lesson 32a,
y
ou’ll notice that when
a
is positive the parabola opens upward. When
a
is negative, it opens
downward.
Graph the following equations:
1.
y
=
x
2
+ 4
x
+ 4
2.
y
=
x
2
- 6
x
+ 8
3.
y
=
x
2
- 6
x
+ 4
direction: up
y
-intercept:
y
= 0+0+4
(0,4)
verte
x
:
x
= −
= −
4
2 1
2
( )
y
= 4 +4(-2)+4 = 0
(-2,0)
s
y
mmetr
y
point: (- 4,4)
4.
y
=
x
2
+ 8
x
+ 15
5.
y
=
-x
2
+ 3
x
- 2
6.
y
=
-x
2
+
x
+ 12
Luttrell 2012
125
Name: ______________________
Date: _____
8L – Quadratic Word Problems
A quadratic equation can be recognized by the shape of the data when plotted. Or ma
y
be an
equation was given and it was in the form
y
=
ax
2
+
bx
+
c
where a ≠0. If you need to e
x
trapolate
or interpolate (predict) some values, it is best to find the equation first. Then using the equation,
the prediction of the needed value should be accurate.
Solve:
1. A ball is kicked so that it lands 50 feet awa
y
3 seconds later. The height of the ball at an
y
given moment is found b
y
h
= -12
x
2
+ 36
x.
According to the equation, what is the ma
x
imum
height the ball travelled?
2. Wind chill temperature is given b
y
the formula
C
w
F
= −
+
1
6
2
, where
C
is the wind chill
temperature,
w
is the wind speed in miles per hour, and
F
is the Fahrenheit temperature in still
air. On Jul
y
12, the wind speed was 15 mph and the still air was 88
°
F. What was the wind chill
temperature for that da
y
? Which variable has greater effect on
C
? E
x
plain
y
our conclusion.
3. Carbon Dio
x
ide emissions were found to be increasing since 1975. The following equation
has been simplified to make calculations easier,
x
represents the number of
y
ears since 1975 and
y
represents CO
2
measured in parts per million: y = 0.005x
2
+ x + 300. What would
y
ou e
x
pect
for the
y
ear 2006?
4. The horsepower required to overcome wind drag on an automobile is appro
x
imated b
y
H = 0.01s
2
+ 0.01s – 0.1
where s is the speed of the car in miles per hour. What is the speed
of the car when the horse power is 2?
Luttrell 2012
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Name: ______________________
Date: _____
8m –Complex Numbers
Imaginary Numbers are comple
x
numbers that are square roots of negative numbers. The
y
are
called imaginar
y
because at the time of their discover
y
, no one could imagine such numbers
e
x
isting. But the
y
do!
Y
ou’ll encounter them in engineering.
The unit for an imaginar
y
number is
i
which is equal to
−
1
. Any imaginar
y
number can be
written as a multiple of
i
. Take
−
4
as an e
x
ample. It can be written as a product,
−
1 4
which can be written as 2
i
. Of course the e
x
ample could be irreducible like
−
=
13
13
i
.
Imaginar
y
numbers can be plotted on the “imaginar
y
” plane. It looks ver
y
similar to the real
number line! Now if
y
ou intersect both the real and imaginar
y
lines at zero.
Y
ou should have the
comple
x
plane. The comple
x
plane contains all numbers. It’s onl
y
a matter of finding them on
the plane. A comple
x
number has the form
a + bi
, where
a
is the real number and
bi
is the
imaginar
y
number.
Y
ou plot the number as if
y
ou would in the cartesian coordinate s
y
stem.
Y
ou
go left/right from the origin as man
y
units as
y
our real number (
a
) and up/down the number of
b
units.
The graphs of 2 + i, 3 - 2i, and 3i are plotted
to the right. (in their approximate spot)
●3i
i
● 2 + i
5
●3-2i
Y
ou can combine comple
x
numbers like an
y
real number, e
x
cept
y
ou can onl
y
add reals together
and imaginaries together. (Like Terms:)
Simplif
y
the following. Graph the results.
1. (-3 - 2
i
) + (-3 + 2
i
)
2. 3 - 2
i -
(3 - 2
i
)
3. 5 -
i+
3
- 3i
4. -3-2
i
-(10-12
i
)
Solve the following and simplify in terms of its complex solution:
5. 4x
2
+8x +9 = 0
6. x
2
+x + 1 = 0
Luttrell 2012
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Name: ______________________
Date: _____
8n –Complex Numbers continued
Multipl
y
ing comple
x
numbers is similar to multipl
y
ing pol
y
nomials. There is just one e
x
tra step
of simplification and that is to remember
i
2
2
1
1
=
−
= −
(
)
.
For e
x
ample: (3 - 2
i
)(2 -
i
) = 6 - 3
i
- 4
i
+ 2
i
2
= 6 - 7
i
- 2 = 4 - 7
i
.
Multipl
y
and simplif
y
:
1. (2 + 3
i
)(3 - 2
i
)
2. (3 + 2
i
)(3 - 2
i
)
3. (1 - 2
i
)(1 - 2
i
) 4. (1-
i
)(2 + 2
i
)
Sometimes it is necessar
y
to know how far a point is from the origin, otherwise called modulus
or magnitude. The magnitude is indicated b
y
vertical bars around the comple
x
number. So | 4 +
3
i
| would be 5. How do
y
ou contrive that? By using the Pythagorean Theorem, use the origin
and given point as vertices of a right triangle.
Evaluate. Graph and show work with triangles.
5. | 3 - 2
i
|
6. |-2 - 2
i
|
7. |1 + 2
i
|
8. |-5 + 12
i
|
What can you do really fast to determine (without actually solving) if the equation has real roots?
If the discriminant (b
2
– 4ac) is negative there are no real solutions. Remember, negative
numbers under a square root are not possible with real numbers.
Comple
x
numbers are commonl
y
found when solving quadratic equations. Solve the following
for its x-intercepts, simplif
y
ing
y
our answers completel
y
.
9.
y
= 3
x
2
- 6
x
+ 4
10.
y
=
x
2
- 3
x
+ 3
11.
y
= 2
x
2
+ 7
x
+ 8
Luttrell 2012
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Chapter 8 Test
Name: ___________________________
Date: ____________
SHOW WORK
. A calculator is allowed on this test. Attach any scratch paper that’s used.
1. Write two examples of a like term for each of the following:
A. 7x
B. -3x
2
y
7
2. Combine the like terms:
A. (x
2
+ 2x - 1) - (2x
2
+ 3x + 3)
B. 5x
2
y
2
- 4x
2
y
2
+ 3xy
3
3. Determine each product:
A. 4(3b)
B. -2p
2
(3p
3
)
4. Determine each product:
A. 2x (2x + 3)
B. -12(3 + 2t)
5. Factor each binomial:
A. 25a + 30a
2
B. 9c
3
- 15c
6. Factor each binomial:
A. a(a + 6) + 7(a + 6)
B. -16(x - 2) + 48(x - 2)
7. Expand (foil):
A. (a + 1)(a - 3)
B. (2x - 3)(3x + 4)
5
5
5
5
5
5
5
Luttrell 2012
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Chapter 8 Test, continued
8. Factor what you can and then reduce each fraction:
A. (6a - 12) ÷ 3
B. (4a
2
+ 12a - 16) ÷ (a + 4)
C. (x
2
+ 7x + 12) ÷ (x + 3)
9. Solve by factoring:
A. a
2
+ 5a – 14 = 0
B. 121 + 22m + m
2
= 0
10. Solve by the quadratic formula:
A. 3a
2
+ 2a – 5 = 0
B. 12 + 7m + m
2
= 0
11. Graph y = 2x
2
– 5x + 3.
12. Use the discriminant to determine how many x-intercepts the graph y = 2x
2
– 5x – 3.
Bonus of three points: (x
2
+ x - 2) ÷ (x
2
- 4x - 12)
⋅
(x
2
- 5x - 6) ÷ (x
2
- 1).
10
5
5
5
5
0
Luttrell 2012
130
Name: ______________________
Date: _____
Algebra Cumulative Review
1. Simplif
y
:
2
15
3
2
3
2
3
2
x
xy
x y
⋅
(
)
.
2. What is the additive inverse to -3?
3. Solve for
p
:
I = prt
.
4. Graph the equation
y
= 2
x
- 3.
5. Graph the solution to |3
x
+ 7| < 8.
6. Transform the given equation into Standard (A
x
+ B
y
= C) form:
5
4
2
3
y
x
=
−
.
7. Find the linear equation perpendicular to
5
4
2
3
y
x
=
−
through the point (0,1).
8. Compare the point-slope formula of a line and the slope formula.
9. Write a linear equation through the points (3,2) and (5,8).
10. E
x
pand: (2
x
-5)
2
.
11. Solve the following b
y
two methods:
0
= 3
x
2
- 5
x
+ 2.
12. Solve the equation for
r
:
A
rr
=
π
2
.
13. Solve the following over the comple
x
plane:
0
= 5
x
2
- 6
x
+ 5.
14. Simplif
y
:
4
5
x
xy
+
.
Luttrell 2012
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Appendix
Algebra II Review Sheets
The appendix contains worksheets that were designed for students taking Algebra II with a
textbook by Paul A. Foerster entitled Second Edition of Algebra and Trigonometry: Functions
and Applications. These were questions were pulled from his book for students seeking extra
questions to practice for a test. You will find these worksheets handy as a test or exam review
whether or not you are using the same textbook.
So note that each sheet in the appendix covers the major themes of an Algebra II course. The
material to build these topics would have to be found in another document.
Luttrell 2012
132
Name: ______________________
Date: _____
1 Sets and Operations of Numbers
Directions: Use your algebra textbook to determine how to do the following questions. Read
carefully the text as well as the examples in the textbook. Try a few questions out of the
textbook for additional practice. Help is provided to clarify any concept.
1. Identify the different sets to which each number belongs.
Type
number
Integers
(Z)
Natural
(N)
Rational
(Q)
Imaginary
Whole
Complex
(C)
0
2.71828...
−
27
3
3.14
6
2. Simplify: (2x-3)(x+5)
3. Simplify: 4x - 2[3x - (x-2x)]
Explicitly solve two equivalent forms for absolute equations (inequalities). Then graph the
solution:
4. 4 - 3x > -5
5. | 3x + 2 | < 5
6. | x + 1 | < -3
7. 2 < | 2x + 3 | ≤ 5
8. | 5x - 3 | = 12
9. 8(x – 2) < 12
10. Expand: (x + y)
3
11. Solve (x + 4)(x - 5)(2x + 3) = 0.
Luttrell 2012
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Name: ______________________
Date: _____
2 Functions and Relations
1. Solve: x
3
= 16x, domain
∈
{x | x > 0}
2. Show the algebra to expressing
5.1121212... as a ratio of integers.
3. Determine if the following is a polynomial. If it is, give its name by both degree and term.
A. x
2
yz
5
- 9x
2
y
3
z
B. 4xy
3
- 7xy
7
+ 5x
2
y
5
C. 3x
2
y
-3
- (2xy
2
)
3
4. Graph: y = -x
2
- 4x.
5. Graph
f x
x
x
x
x
( )
,
,
=
≤
−
>
2
4
3
4
What is the range?
What is the range? What is f(4)? f(0)?
6. Explain or show the different ways you can determine if 3x
2
- 3y
2
- 6x + 5 = 0 is a function.
Is it a function?
7. Name the axiom(s):
a. 12(4×3) = (12×4)(3) ______________
b. 3+(-3) = 3 - 3 = 0 ______________ ____________
c. 2(½) = 1
______________
d. 3×4×2 = 2×3×4
______________
8. Fill in the justifications for each step of the proof:
Multiplication Property of Zero
Step
Justification
0 = 0
0 + 0 = 0
x(0+0) = x(0)
x(0+0) = 0 + x(0)
x(0)+x(0) = 0 + x(0)
x(0) = 0
Luttrell 2012
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Name: ______________________
Date: _____
3 - Linear Functions
1. Graph y + 4 = -2(x - 1).
2. Graph 4y - x = 8.
3. Find the y-intercept and slope of
4. Transform y - 3 = ¾(x + 5)
3x - y + 5 = 0.
into Ax + By = C form.
5. Graph the plane: x - 3y + 2z = 6.
6. Determine the type of system: dependent,
Label the intercepts and axes.
inconsistent, or independent.
2x - 3y = 1
-5x + 7.5y = 4
7. Write the linear equation perpendicular
8. Write the equation of the line
to 2x + 4y = 5 that passes through (3,1).
parallel to 2x + 4y = 5 that passes
through (3,1).
Luttrell 2012
135
Name: ______________________
Date: _____
4- Systems of Linear Functions
Solve the following systems by each of the following methods: Substitution, Elimination,
Determinants (Cramer’s Rule), Augmented Matrices, Graphing, Inverse Matrices. Attach any
additional sheets of paper you used to complete the different methods.
1. 2x + y = 3
2. 7x + 3y = 6
3. x – 5y = -6
x - 4y = -3
5x - 2y = 25
3x – 6y = 0
Solve each of the following systems by a different method: substitution, elimination,
determinants, and augmented matrices.
4. 2x - y - z = 6
5. 3x + 2y - z = 3
x + 5y + 3z = -10 x + 4y - 4z = 7
3x - y - 5z = 4
-2x - 3y + 5z = 0
Luttrell 2012
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Name: ______________________
Date: _____
5 - Quadratic Functions
Convert the quadratic equations into Vertex Form. Graph.
1.
y
= 3
x
2
- 12
x
+ 5
2.
y
= -2
x
2
+ 8
x
- 3
3.
y
> 5
x
2
+ 10
x
+ 12
Verify the number of solutions by using the discriminant.
4. 0 = 3
x
2
+ 4
x
+ 5
5. 2
x
2
- 3
x
= 3
6.
x
2
+ 2
x
= 5 - x
7. Write the quadratic equation that passes through (-3, 37), (1,1), and (2,7).
For #8- #10,use
f(x)
= 3
x
2
+
x
- 2 to find
x
such that the equation is true.
8.
f(x)
= 0
9.
f(x)
= 3
10.
f
(5
)
=
x
Luttrell 2012
137
Name: ______________________
Date: _____
6a - Exponentials
Simplify. No complex number should remain in the denominator!
1. (3
x
- 2)
2
2. -7
2
3.
(
)
(
)
−
−
2
6
3
2
3
3
2
x y
xy
4.
(
)
(
)
9
9
35 10
32 11
5.
(
)
(
)
1001
77
4
6
7
x
x y
−
−
÷
6.
(
)
36
2
3
2
x
7.
(
)
128
4
2
7
x
8.
(
)
3
6
3
3
5
x
x
9.
x
y
12
8
4
10.
5
2
1
i
−
11.
2
1
4
+
−
i
i
12.
8
4
2
4
9
i
i
− +
Luttrell 2012
138
Name: ______________________
Date: _____
6b - Logarithmic Functions
1. What is the definition of a logarithm?
Simplify into one logarithm:
2. log 2 + log 4 - log 3
3. 2log 4 - log 5 + 3log 3
Expand into several logarithms:
4.
log(
)
x y
z
2
3
5.
log(
)
(
)
x
x
2
1
2
+
6.
log(
)
x
x
2
+
Change into a natural logarithm:
7. log 5
8.
log
2
6
9.
log
3
7
Simplify:
10.
5
6
64
64
log
11.
10
2005
log
12.
log
8
2
13. log 1
14. log 0
15. log 1000
16.
log
4
64
17.
2
3
27
log
18. log 2 + log 5
Luttrell 2012
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Name: ______________________
Date: _____
6c - Solving Logarithmic Functions
Solve exactly:
1.
4
64
x
=
2.
2
8
1
x
+
=
3.
3
25
x
=
4.
10
144
x
=
Solve simultaneously these systems in questions 5-6.
5.
y
y
x
x
= ⋅
= ⋅
3 2
6 2
3
6.
y
y
x
x
=
= ⋅
3
2 3
2
7.
log (
)
log (
)
2
2
4
2
3
x
x
−
+
−
=
8.
log (
)
log
3
3
2
1
x
x
−
+
=
9. What are properties of inverses? How do you find an inverse to a function?
10. Graph and determine if
y
x
= −
5
and
y
x
= −
log
5
are inverses.
Find the inverse of:
11.
f x
x
( )
=
+
3
2
2
12.
y
x
=
3
13.
f x
x
( )
(
)
=
−
1
2
Determine if the following pairs of equations are inverses:
14.
f x
x
g x
x
( )
( )
(
)
=
+
=
−
2
2
2
2
15.
f x
g x
x
x
( )
( )
log
= ⋅
=
2 3
2
3
Luttrell 2012
140
Name: ______________________
Date: _____
7 - Rational Functions
Expand these polynomials:
1. (6
x
+ 5)
2
2. (
x
-
y
)
3
3. (2
x
+ 4)
2
Factor these polynomials:
4.
x
2
- 6
x
- 27
5. 12
x
2
+ 25
x
+ 12
6. 7
a
4
- 28
b
2
7. 9
x
2
- 4
y
6
8. 8
x
3
- 27
y
12
9. 4
xy
-3
y
-8
xy
2
+ 6
y
2
Simplify these rational expressions:
10.
x
xy
y
x
y
y
x
y
x
2
2
2
2
2
+
+
−
÷
+
−
11.
1
1
4
1
2
1
−
−
+
−
x
x
12.
x
x
x
x
x
+
−
+
+
+ +
2
1
1
1
3
2
Solve these equations:
13. 2
x
2
- 5
x
= 7
14.
x
x
x
x
+
−
=
−
3
2
3
18
4
9
2
15.
2
1
3
1
5
x
x
+
−
−
=
16. A rectangular piece of cardboard has area of 200 cm
2
. The length is four times its width.
Find the dimensions of the cardboard.
17. A metal worker wants to make sure an open box made from 6" by 8" sheet of metal has
maximum volume. The box was made by cutting out equal squares from the corners of the
sheet and then bending edges. Once you determine the equation to represent volume, use
a calculator to find the maximum volume and corresponding dimensions of the box. Explain
how you got your answer.
Reduce the equations. Graph, labelling the intercepts and discontinuities. Differentiate between
removable and nonremovable asymptotes. No calculator!
18.
f x
x
x
x
x
( )
(
)(
)
=
− −
+
+
2
2
6
2
3
19.
f x
x
x
( )
=
+
+
3
3
1
3
