Download Algebra 2

Honors Algebra 2 – Midterm Review – Part I
P.1 Review of Real Numbers & Their Properties
1)
Draw a diagram that illustrates the relationship between the following number sets:
real numbers, integers, whole numbers, rational numbers, natural numbers, imaginary
numbers, pure imaginary numbers, and complex numbers.
2)
Use the above diagram to name the subsets
of the following belong.
a) -0.05
b)
c) 2/3
d)
e) -15
f)
g) 0
h)
of the complex numbers to which each
0.666…
4 + 8i
-5i
2 7
3)
Show that each decimal can be written as a rational number.
a) 0.32
b) 1.125
c) 0.3535….
d) 2.236
4)
State the name of the property for each example.
1
a) (x + 2) · 1 = (x + 2)
b) 7 ·
=1
7
c) 2x + 5 = 5 + 2x
d) 7a(b + 3c) = 7ab + 21ac
e) -29  (4  x) = (-29  4)  x
f) (2x – 1) + 0 = 2x – 1
5)
Use interval notation to describe the given set.
a) x < 4
b) -3 < x < 0
c) x is nonnegative
d) x is no more than 21
e) x is at least 5 units from 0
f) x is at least -11
GD/08
1
P.2 Exponents & Radicals
1)
Simplify each expression involving exponents.
a) (-3a6bc-3)(-5a-7b1)
b) (-2xy3z10)5
c) (3a)-2(a8b0)
d) (7ky4)(-2k2y-2)3
8x -1y 4
-3x 3y 7
e)
f)
(4x )2 y -9
9xy -3
æ 5a -3b 4 ö
ç
÷
8
è 7a b ø
g)
2)
-7xy 3
6xy -3
·
3x 5
21x -2y 4
h)
Simplify each of the following using exponent rules (positive exponents only).
-
2
3
1
4
6x y
a)
1
2
(2x y
-3
æ
8
2
x
e) ç
-4
ç
ç 3y 9
è
3)
-2
1
3 -3
b)
)
ö
÷
÷
÷
ø
(16a
b -6c
4
(7
(2
e)
3
4
i)
3
13
)
2
)(
6 -3 5+ 6
40a 7b 3c 15
k) 5 45 + 6 80
m)
o)
GD/08
c) (4x y z 4 )-3
1
2 4
f) ( -2x y ) ( -3x
5
6+ 2
g)
)
1
6
3
Simplify each radical expression.
a)
12 - 5 24 - 11 75
c)
3
4
3
0 2
3
32x 2 · 3 4x 4
3
54x 2
3
2x
)
-7
5
y )
8 · 5 63
d)
18 · 12
3
2 -2
ö
÷
÷
÷
ø
-3
2 5
8
4- 3
9-2 3
f)
h)
j)
-
(4x y )
( -2x 5y 0 )-3
-5
æ
3x 2
ç
g)
ç -2
ç y9
è
1
6
b)
d)
5
4
256x 6y 8z 10
4
l) 4 3 24 - 33 40 + 150
n)
p)
(3 - 2 )(1 - 2 6 )
3
4
3
16a 5
3b
2
P.3 Polynomials & Special Products
1)
Put each polynomial in standard form, state the degree of the polynomial, and state
the leading coefficient.
a)
2)
3a2 – 6a + 14a5 – a3
b)
7x2y2 – 8xy3 – 11x3y
Simplify and state the degree of the resulting polynomial.
a)
(11x8 – 6x7 + 2x) + (9x7 + 2x8 –3)
b)
(11x2 - 3x – 14) – (6x2 – 5x – 7)
c)
(3x – 7)(5x + 3)
d)
(3a + 2b)2
e)
(2x – 5)(3x2 + 4x – 7)
f)
(x – 4)3
P.4 Factoring Polynomials
Factor completely.
a)
x2 - 7x + 6
b)
x3 - 8
c)
4y3 + 108
d)
3x3 - 24x2 + 21x
e)
8x3 + 27
f)
4x2 + 12x + 9
g)
x2 - 81
h)
64x3 - 27
i)
15x3 + 10x2 + 6x + 4
j)
k3 + 4k2 – 9k – 36
GD/08
3
P.5
Rational Expressions
1)
Perform the indicated operations.
3x 2y
24xy 5
a)
·
16x 3y 4 9x 2y 2
c)
2)
4)
5)
x 3 - 27 2x 3 + 6x 2 + 18x
¸
4x 2 - 25
2x 2 - x - 10
b)
d)
x 2 + x - 6 3x 2 + 6x
·
2x 2 + 9x + 9
x2 - 4
2y 2 - y
4y 2 - 1 (2y - 1)2
·
¸
2y 2 - y - 1 y - y 2
2y - 2
Identify the least common multiple for each of the following.
a) x2 - 9, x - 3, 3
b) x + 5, x2, x
c) 4x, x3, x - 1
d) 1, 3x, x(x + 1)
e) x2, 2x(x - 2), x2 - 4
f) x - 4, x2 - 16, x
Perform the indicated operation.
5x
7
8x
x +2
a)
b)
+
2
x - 4 9x + 18
x -3 x +1
Simplify each of the following.
4
3
1
+2
+
2
x
a) x
b) x
1
4
-8
22x
5x
P.6 Errors and Algebra of Calculus -
c)
8
2x
7 -x
+
x 3x + 3 x + 1
4
3
+
c) x + 2 x + 1
2
8
+
x +2 x +1
Review Algebraic Errors on pgs 51-52
P.7 The Rectangular Coordinate System & Graphs
1) Find the distance between the given points and the midpoint of the segment
joining the points.
a) (-4, 10), (4, -5)
b) (-7, -4), (2, 8)
2) Show that the indicated points form the vertices of the indicated polygon.
a) Right Isosceles Triangle
GD/08
(-4, 2), (6, 2), (1, -3)
4
b) Rhombus (-4, 2), (0, 4), (4, 2), (0, 0)
Honors Algebra 2 – Midterm Review – Part II
1.2 Linear Equations in One Variable
Solve.
a) 3(5 - a) – a = -4(a - 4)
3
2
5
c)
x-7= x+
4
3
6
e) A = P + Prt for t
g) S = L - rL for L
b)
d)
f)
h)
15(x - 2) = -13(x + 1) + 11
x + 2 2x - 1 10 + x
+
=
4
3
6
2
V = r h for h
6x – 3b = 12ax for x
i)
1
3
4
+
= 2
x -2 x +3 x +x -6
j)
5
3x + 1
=
x -2 x -1
k)
x +5
x
+
=5
x -2 x -6
l)
5 2 1 5
- = +
2x 3 x 6
1.3
Modeling with Linear Equations – Review problems from Pgs 107-108
1.4
Quadratic Equations and Applications
Solve each quadratic equation.
1) 3x2 - 24 = 0
3) x2 - 10x - 4 = 0
5) x2 + 5x + 7 = 0
1 2
7)
x + 1 = 33
4
2)
4)
6)
5x2 + 19x = 125 + 19x
-2x2 + 3x -7 = -9
3x2 + 12 = 2(x2 – 3)
8)
7x(1 - x) = 5(x – 1)
Use the discriminant to determine the nature of the roots for each quadratic equation.
Equation
Value of the
Number of Roots
Real or
Discriminant
Imaginary Roots
2
9) 8x – x + 1 = 0
10) x2 – 3 = -6x
11) -12x2 -10x = 4
12) 9x2 – 30x + 25 = 0
GD/08
5
See Pgs 121-124 for applications of quadratic equations
1.5
Complex Numbers
1) Simplify each expression.
a) - 150
b) -256
d)
e) (7i
-18 · -12
c)
13 3
)
-8 + -18 + 5 9
f)
2)
Give the conjugate of each number.
a) 2 + 7i
b)
3)
Solve for x and y.
a) 5x + 2yi = 15 + 4i
b) 3x – 7yi = 2 + -14i
4)
Simplify each expression involving complex numbers.
a) (2 + 9i) + (4 - 11i)
b) (2 + 3i) – (8 + 2i)
2
c) i (4 + i)
d) 8i(1 + 2i)
e) (3 + 4i)(2 – 5i)
f) (12 – 5i)(12 + 5i)
g) (4 – i)²
h) i267
i)
i15
j) (3i)2(8i)
5
26
k)
l)
6i
- 12i
5
3 - 4i
m)
n)
7 - 2i
2 + 5i
3 – 5i
1.6 Other Types of Equations
Solve each of the following (if possible). In your
3
a)
b)
(x + 5 ) - 8(4 - x) = 7x - 3(2 + x)
2
2
2
1
c)
d)
=
+
2
x - 6x + 8 x - 2 x - 4
work, indicate any extraneous solutions.
2x + 1 4x + 2
=
3x - 5 6x - 4
4x2 – 18 = 0
e)
9x2 + 3x + 1 = 0
f)
x4 - 3x2 – 18 = 0
g)
5x4 + 15x3 – 20x2 – 60x = 0
h)
3 - 11x - x = 7
i)
(4 + 5x)3/2 = 125
j)
3
k)
8x + 7 = 15
l)
x2 - 2x = 3x - 6
GD/08
3x + 2 - 3 x - 9 = 0
6
1.7 Linear Inequalities in One Variable
Solve each inequality and write the solution set in interval notation.
a) -63 < 41 – 2x ≤ 17
b) ½ (3 – x) > ¼ (2 – 3x)
c)
4x + 9 < 15
d) 3x - 7 ³ 18
f) 2x - 7 £ 5 and 2x + 6 ³ -12
e)
1.8 Other Types of Inequalities
Solve each inequality and write the solution set in interval notation.
a) (x + 2)2 < 25
b) x2 + 4x + 4 > 9
c) x3 + 2x2 – 9x – 2 > -20
d) 2x3 – x4 < 0
e)
f)
GD/08
7
Honors Algebra 2 – Midterm Review – Part III
2.1
2.2
Linear Equations in Two Variables
-2
and contains the points (3, -4) and (6, y). What is y?
3
1)
A line has a slope of
2)
In 1990 Marc earned $42,360 per year, and he now earns $61,800. What is the
rate of change for Marc’s salary per year?
3)
Write a linear equation using the given information.
-2
a) slope of
and a y-intercept of 4. (Write in standard form.)
3
b) through the points (-6, -1) and (3, 2).
c) through the points (4, 3) and (0, -5). (Write in general form.)
d) horizontal line through the point (3, -7).
e) vertical line through the point (-2, -4).
f) through the point (3, -2) and perpendicular to 2x – 3y = 6.
Functions
1) Determine which of the following relations are functions.
a)
b)
-4
7
d)
{(-3, 8), (0, -5), (2, -6), (3, -5)}
GD/08
e)
c)
6
-3
0
f)
x3 – y2 = 7
y=
7x - 8
3x2 - 11x3
8
2) Identify the domain of each of the following functions.
a) f(x) =
c)
h(x) =
3
9 - x2
b)
g(x) =
2x - 7
d)
k(x) =
6x - 7
2x - 6x2 - 80x
3
x+2
x - 11
3) Given the functions f(x), g(x), and h(x), answer the following questions.
f(x) = - x2 – 7x + 3
g(x) = 9 + 4x
h(x) = 3x + 1
a) Evaluate h(f(g(4)))
ìï -x2 + 3x
4) Given f(x) = í
ïî 7 - 4x
a) f(-5)
1.1
b)
Evaluate f(h(-1))
x £ -2
, evaluate each of the following.
x>5
b) f(11)
c) f(5)
Graphs of Equations
1) Find the x- and y-intercepts of the graph of each equation.
a) y = (x – 3)2
b) y =
c) y = ½ x – 5
2) Determine algebraically if the graphs for the given equations have x-axis symmetry,
y-axis symmetry, or origin symmetry.
a)
b) x – y2 = 1
c) y = x2 – 3
3) Write the standard form of the equation of the circle with the given information.
a) center (-6, 2) with radius 4
b) center (5, -3) with solution point (2, -1)
GD/08
9
c) endpoints of the diameter (-2, -3) and (4, -10 )
2.3
Analyzing Graphs of Functions
1) Given the functions f(x) = 3x4 – 4x2 + 9,
g(x) = 7 – 2x, and
h(x) = x3 – 5x
a) Determine if f(x) is even, odd, or neither.
b) Determine if g(x) is even, odd, or neither.
c) Determine if h(x) is even, odd, or neither.
2) Using the graph shown, answer the following questions.
a)
b)
c)
d)
e)
f)
Domain of f(x)
Range of f(x)
f(2) =
g(-1) =
f(g(0)) =
Determine the intervals over
which f(x) is increasing, decreasing,
or constant.
3) Find the zeros of each function.
a)
f(x) = 3x2 + 13x - 10
b) g(x) = x3 + 3x2 – 16x - 48
c) h(x) =
d)
f(x) = 9x4 – 25x2
e) k(x) =
f) j(x) =
4) Evaluate the function for the given value of x.
a) f(x) = x2 – 10x
f(4)
f(-8)
f(x – 4)
b) f(x) =
GD/08
10
f(12)
2.4
f(40)
f(
)
Parent Functions
Match each function with its name.
1) f(x) = [[x]]
a) cubic function
2) f(x) = x2
b) square root function
3) f(x) = |x|
c) identity function
4) f(x) = c
d) linear function
5) f(x) = x
e) greatest integer function
6) f(x) = x3
f) squaring function
7) f(x) = ax + b
g) reciprocal function
8) f(x) =
h) absolute value function
x
9) f(x) = 1
x
i) constant function
2) Write the linear function f such that it has the indicated function values.
a) f(-5) = -1, f(5) = 1
b) f( ½ ) = -6, f(-4) = -3
3) Evaluate the function for the indicated values.
f(x) = [[x + 3]]
a) f(-2) =
GD/08
b) f( ½ ) =
c) f(-21.6) =
11
4) Graph the function.
a)
b)
5) Write the given absolute value functions as piecewise-defined functions.
a) f(x) = |2x – 8|
b) f(x) = |1 – ½x|
6) Write a piecewise-defined function for each graph.
(omit for now – these were drawn on)
a)
GD/08
b)
12
2.5
Transformations of Functions
Function
Parent
Function
Domain
Range
Sketch
f(x) = x + 4 - 1
2
f(x) = (x - 3) + 2
GD/08
13
f(x) = - x - 1 + 3
3
f(x) = (x + 2) + 4
f(x) = [[x + 1]]
GD/08
14
1) Use the graph of f to sketch each graph.
a) y = f(x) – 2
b) y = f(-x)
2) Identify the parent function and use function notation to write g in terms of f.
a) g(x) = | x – 7 |
GD/08
f(x) = ______
g(x) = _________________
b) g(x) = (x + 4)3 – 1
f(x) = ______
g(x) = _________________
c) g(x) = -(x + 5)2 + 6
f(x) = ______
g(x) = _________________
d) g(x) =
f(x) = ______
g(x) = _________________
15
2.6
Combinations of Functions
1) If f(x) = x + 3 and g(x) = 4x – 1, find the following:
a) f(x) – g(x)
b) f(x)  g(x)
c) f(g(x))
d) g(f(x))
2) Find (f g)(x) and (g f)(x). Find the domain of each function and each composite
functions.
a) f(x) = (1/3)x – 3,
g(x) = 3x + 1
2.7
b) f(x) = x3 – 4,
g(x) =
c) f(x) = 2x2,
g(x) =
Inverse Functions
1) Find the inverse function of f. State the domain and range of f and f-1.
a) f(x) = ½x – 3
c) f(x) = x3 + 2
b) f(x) =
2) Determine if the function is one-to-one and has an inverse.
a) {(-3, 1), (2, 5), (4, 5), (-8, 7)}
c) f(x) = 4
e)
GD/08
b) {(-6, 4), (3, 0), (7, 1), (6, 2)}
d)
y=
2
x-3
f)
16