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Review Packet - Algebra 234 Final Exam 2015
I. NON - CALCULATOR SECTION
Unit 2.6 - 2.7
Factor: completely:
1.
3a2 - 24a + 48
4.
6p2 + p -12
2. 4n 2 - 32n + 48
5.
16p2 - 64
6.
Solving Polynomials:
Solve the following polynomials. (Find the value(s) of x)
1.
3. 6x 2 +13x - 5
2.
3.
4.
5.
6.
6x 3 =8x 2 +8x
f (x) = 10x 3 - 81x 2 + 71x + 42 Given that 7 is a root, find the other roots.
f (x) = 3x 3 + 34x 2 + 72x - 64 Given that f(-4)=0, find the other roots of the function.
7.
f (x) = x 3 + 5x 2 -18x - 72 Given that x-4 is a factor, factor the polynomial completely.
8. What is the value of the function
when x = 1?
2.6b The Quadratic Formula - Solve the following quadratic equations using the quadratic
formula.
Quadratic Formula: _________________________ Discriminant:__________________
Find the discriminant. Determine the number and type of solutions.
1.
Discriminant: ____
Number of Solutions: ______ Type of solutions: ________________
Use the quadratic formula to solve for the x-intercepts:
2.
3.
2.6c Solving Quadratics using the Square Root Method
Solve:
1.
2.
2.6d Completing the Square
Solve by completing the square(magic hat method):
1.
2.
2.7 Comparing Quadratics
1. Non-calculator. Which function has a smaller minimum value? Explain how you know?
a.
b.
2. Calculator:
Suppose Mark and Steven each throw a baseball into the air. The height of Steven's baseball is
given by
where h is in feet and t is in seconds. The height of Mark's baseball is given by the graph below.
A. How long did it take for Mark’s baseball to reach its maximum height? How long did it take for
Steven’s baseball to reach its maximum height?
Mark:
Steve:
B. Approximate the maximum height of Mark’s baseball. Calculate the maximum height of
Steven’s baseball.
Mark:
Steve:
C. Approximate how long it took for Mark’s baseball to hit the ground? Calculate how long it
took for Steven’s baseball to hit the ground?
Mark:
Steve:
Unit 3 Functions
1. State if the set of ordered pairs is a function using the graph and explain why. Then state the
domain and range.
Domain:
Range:
Explain:
2. State if the set of ordered pairs is a function using a mapping diagram and explain why. Then state
the domain and range.
(0, 1) (-2,3) (4, 0) (-2, 7)
Domain:
Range:
Explain:
3. Given the table below. Identify the domain and range. Determine if the table represents a
function.
x
-1
-2
0
1
-2
y
4
5
3
7
9
Domain:______________________________________
Range:______________________________________
Does the table represent a function?
Circle: Yes or No
Explain your reasoning.
_____________________________________________________________________________________
___________________________________________________________________________________________________________________
4.
For the following functions:
f (x) = x 2 - 4, g(x) = x - 4,h(x) = x + 2
a.
Find g(-8).
c.
When will f(x) = 0?
5.
b.
Find f(7)- h(2).
f (x) = (x - 2)2 find f(-3) + f(½).
Given the functions f (x) = 4x 2 + 8x - 2 , g(x) = 3- x , and h(x) = -3x + 5 find:
6.
7.
f (-1)
g(1)
Given the functions f (x) = 4x 2 + 8x - 2 , g(x) = 3- x , and h(x) = -3x + 5 find:
8.
10.
f (g(2))
9.
h(x) – g(x)
g(h(x))
For the set of numbers, graph the set on a number line and write it in interval notation.
10.
all real numbers between -5 and 2, including -5 but not including 2
11.
Write in set builder notation
12.
Find the domain and range of each relation.
a.
b.
Domain:
Domain:
Range:
Range:
13.
From the different types of functions we learned, name the type of function underneath its
graph. Be specific when naming each function.
Function Word Bank: Linear, Quadratic, Cubic, Absolute Value, Logarithmic, Exponential,
Square Root, Cube root, Piecewise, and Rational
a.
b.
________________________
__________________________________
d.
c.
_________________________________
__________________________________
Unit 4 Interpretations of Functions
4.1 Inverses
1. Find the inverse of
2.
. Show all steps.
Find the inverse of f(x) = 3x-6. Show all steps.
4.2 Exponential Equations
Solve for each variable:
1.
2.
4x- 2 =
1
8
3. 4
x+ 2
=16
2x =
4.
1
16
4.2 Logarithmic Functions
Solve each:
1.
log 2 x = 5
4.
7.
log 3
1
=x
81
log8 4 = x
2.
5.
log 3 9 = x
1
2
log 2 8 = x
8.
3.
log x 4 =
6.
log 9 x = - 12
1
2
log 7 1= x
4.3 Rational Expressions
A. Simplifying using Multiplication and Division
(3xy )
3 2
1.
4
9x y
3
2
æ 2a ö æç b ö
•
è ø è8ø
4. b
x + 9x + 20 x + 3
•
x 2 + 6x + 9 x + 4
2.
x 2 - 5x - 4
x 2 - 16
x 2 +11x + 28
2x 2 + 8x
3.
x 2 + 5x + 4 2x + 2
•
2
5. x + 2x + 1 x + 4
2
6.
7.
6x 3 3xy
2 ¸
y
2
8.
ab 3 10a 2 b
2 ¸
2c
3c
10.
4x 2 + x - 3 4x 2 -11x + 6
¸
x 2 + 2x +1
x 2 -1
12.
d 2 + 6d + 9
d2 - 9
¸ 2
d 2 - 25
d - 3d -10
9.
11.
u2 - 9 u + 3
¸
u + 2 u2 - 4
x2 - 9
x 2 + 5x + 6
¸ 2
x 2 - 5x + 6
x -4
B. Solving Rational Equations
x2 x + 2
=
2
1. 9
3.
5.
x+
2.
2x
3x - 2
=
x-2 x -2
3x - 7 =
6
x
4.
7x + 29 =
-30
x
2x
1
2
- =
x - 3 2 2x - 6
Complex Fractions
1.
2.
3.
4.
5.
Trigonometry Mini Unit
1. Write each ratio as simplified fraction.
II. Graphing Calculator Section - Round to the nearest hundredth when
applicable
1.
2.
3
2
Find the x-intercepts of f (x) = 4x - 2x -10x + 6 with your graphing calculator.
3
2
Find all solutions with your graphing calculator of h(x) = 5x + 4x -16x - 2 .
Graph the following on a graphing calculator and sketch the graph. Identify key characteristics. Write the
domain and range in interval notation.
3.
f (x) = x - 4
Domain:
Sketch:
______________________
Range: __________________________
x-intercept: _____________________
y-intercept: __________________________
4.
f (x) =
Domain:
5
x-4
Sketch:
______________________
Range: __________________________
Fill in the blanks:
Asymptotes: VA x = ________ (fill in the blank)
HA y = ________ (fill in the blank)
5.
h(x) = 4 x
Domain:
Sketch:
______________________
Range: __________________________
Asymptote: y = _______ (fill in the blank)
6.
f (x) = x -1
Domain:
Sketch
______________________
Range: __________________________
x-intercept: _____________________
y-intercept: _______________________
III. Problem Solving
1. The volume of a box is
(Non Calculator)
. Find the expression for the missing dimension.
x-4
2x + 1
2. Find the simplified expression for the area of the rectangle below.( Non Calculator)
x6
3. If log 3 x = 2 and log 4 y = 0 , find the numerical value of y - x in simplest form.
4. The period of a pendulum is the time it takes to swing back and forth. The period, t (in seconds of
a pendulum of length d (in inches) is given by d = 9.78t 2 . The longest pendulum, in the world is
part of a clock on a building in Tokyo, Japan. This pendulum is about 888 inches long. What is the
period of this pendulum? Round one decimal place.
5. Chris jumped off a diving board into a swimming pool in while vacationing with his family. His
height as a function of time could be modeled by h(t) = -16t 2 +10t +15 , where t is the time in
seconds and h is the height in feet. How long did it take for Chris to splash down in the pool?
6. If a 15 foot ladder is leaned up against a wall. It makes a 48 angle with the ground.
How many feet up the wall will it reach? _______
Round to the nearest tenth. Include a model sketch in your solution.
Model:
Equation and Math Work:
7. It’s Summer time, which means that it is time to go shopping for Summer Clothes. You want to
buy a pair of sandals, but only have $25. You become very excited when you get an email stating that
Famous Footwear is having a 30% off sale. You are also a member of Famous Footwear Rewards
program so you receive an additional $5 off your purchase.
Let x represent the regular price of the sandals.
a. Write a function f(x) that represents the cost of the sandals after the discount of 30%.
b.
f(x)=
c. Write a function h(x) that represents the cost of the sandals after the $5 off your purchase.
h(x) =
d. Write a function, k(x), that represents how much you would pay if you use the discount first
followed by applying your rewards.
k(x) =
e. Write a function, g(x), that represents how much you would pay if you use the rewards first,
then apply the discount.
g(x) =
f.
The sandals you want to purchase cost $45. How much would you pay for them using both
functions k(x) and g(x)? Show all work step by step.
Calculations for k(x):
Calculations for g(x):