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Algebra Midterm Study Guide
*Remember - this is not everything you need to know. You will have to refer back to old packets, study
guides, and do practice questions in order to be successful on the midterm exam. :)
Calculator Keys
Negative
Fraction
Converting Fractions to Decimals/Decimals to Fractions
Absolute Value
Exponent
Graphing
Table
Graphing Picture
Fixing your Graph
Test Taking Strategies
Multiple Choice Strategies
Free Response Strategies
Compare your Choices
Eliminate Choices
Guess and Check
Multiple Methods of Solution
Process of Elimination
Use your Calculator
Read carefully
Highlight keywords
Break the question into chunks that you understand.
Show all work!
Use your calculator
1
“Catch Phrases” for Free Response Questions
If the question asks you to
explain...
You answer with...
Is it rational?
“Yes, because it’s an integer”
Is it irrational?
“Yes, because it’s a non-terminating,
non-repeating decimal”
Is the point a solution?
“Yes, because it lies on the line”
“Yes, because it lies in the shaded region”
OR
“No, because it does not
lie on the line”
“No, because it does not
lie in the shaded
region”
What does the slope mean?
“Rate of _______ per ________”
What does the Y-intercept mean?
“Initial amount, cost, fee…”
Is the point a solution? (system of
inequalities)
“Yes, because it lies in the double shaded
region”
OR
“No, because it does not
lie in the double
shaded region”
Is it a function?
“Yes, because each input has only one
output”
OR
“No, because an input has 2 different outputs”
Exponential growth and/or decay (keywords,
see page 9)
“Initial amount of _______”
“Factor of growth/decay”
Is it a linear function?
“Yes, because there is a constant rate of
change”
Is is an exponential Function?
“Yes, because there is not
a constant rate of
change”
2
Unit 1 - Polynomials
Adding/Subtracting Polynomials
Multiplying Polynomials/Distributive Property
(Box Method)
1. Identify terms that have the same base and same
exponent.
2. Add coefficients.
3. Keep the base and the exponent.
4. Standard form - greatest exponent first and then
follow in descending order.
1. Multiply coefficients.
2. Keep the base.
3. Add the exponents.
4. Combine like terms (see box to the left).
Example of Substitution
Example of using the “Box Method”
3
Unit 2 - Equations
Example of Solving an Equation
Example of Solving a Literal Equation
Given the formula , solve for t.atd =
2
1
2
4
Unit 3 - Inequalities
*When solving inequalities, we use the same steps as solving equations.*
**Remember to reverse the inequality symbol when we divide by a negative number!!**
x
< 4
x
> 4
x ≤ 4
x ≥ 4
x
is less than 4”
Open circle
Shade left
x
is greater than 4”
Open circle
Shade right
x
is less than or equal to 4”
Closed circle
Shade left
x
is greater than or equal to 4”
Closed Circle
Shade right
Interval Notation
Meaning
Compound Inequality
Graph
(-2, 3)
-2 is not included
3 is not included
{-1, 0, 1, 2}
-2 < x
< 3
(-2, 3]
-2 is not included
3 is included
{-1, 0, 1, 2, 3}
-2 < x
3
[-2, 3)
-2 is included
3 is not included
{-2, -1, 0, 1, 2}
-2 x
< 3
[-2, 3]
-2 is included
3 is included
{-2, -1, 0, 1, 2, 3}
-2 x
3
“at most”
“no more than”
“at least”
“no less than”
5
Unit 4 - Linear Equations and Inequalities
Graphing Linear Equations
Standard form of a line:
y
= m
x
+ b
m
represents the slope/rate of change of the line.
b
represents the y
-intercept of the line.
x
and y
represent a point, (x
, y
), on the line.
Key Feature
Algebraic Solution
Graphic Solution
x-intercept
Cover the y
and solve for x
.
Point where the function intersects with the x
-axis.
y-intercept
Cover the x
and solve for y
.
Point where the function intersects with the y
-axis.
Solution
Substitute (x
, y
) in your equation.
If true, then it is a solution.
If false, then it is not a solution.
In order for a point to be a solution it must be on the
table and line.
Rate of Change
x x
2 1
y y
2 1
“Rise over Run”
Graphing Inequalities
Notes
Example
< is a dashed line and shade down.
> is a dashed line and shade up.
is a solid line and shade down.
is a solid line and shade up.
*Remember - you must have...
A Table
Arrows
Labels
...to earn full credit on your midterm exam!
6
Unit 5 - Systems of Linear Equations & Inequalities
Linear System
Inequality System
Graph each line separately!
The point of intersection is your answer. This
is called your “Solution.”
Remember - you must have tables, arrows,
and labels for everything in order to get full
credit on your midterm!
Graph each inequality separately!
< is a dashed line and shade down.
> is a dashed line and shade up.
is a solid line and shade down.
is a solid line and shade up.
The double shaded section is your “Solution.”
Any point in this area (not on the lines) is true
for BOTH inequalities.
Remember - you must have tables, arrows, and
labels for everything to get full credit on your
midterm!
Solving Systems of Equations Algebraically
Elimination Method
Substitution Method
Step 1: Switch your coefficients and make one
negative (multiply top on bottom, and bottom on
top).
Step 2: Distribute to each term.
Step 3: Combine like terms and solve for y
.
Step 4: Substitute y
into one equation and solve
for x
.
*Make sure equations are in “y
” = format.
Step 1: Set equations equal to each other.
Step 2: Solve for x
.
Step 3: Substitute x
into one equation and solve
for y
.
Your solution should be written as a point (x
, y
)
7
Your solution should be written as a point (x
, y
)
unless it is a real world situation.
unless it is a real world situation.
Unit 6 - Functions
Key Feature
Meaning
Example
Domain
“Input” of a function (x
-values).
Domain: All Real Numbers
Range: (x)f ≥ 2
Range
“Output” of a function (y
-values).
*Each input can only have 1 output.*
(see page 2 for free - response catchphrases)
8
Unit 7 - Exponential Growth & Exponential Decay
Exponential Growth Formula
(Memorize!)
Exponential Decay Formula
(Memorize!)
(t) (1 )f = a + r
t
a
= initial amount
r
= rate, written as a percent
t
= time
(t) (1 )f = a r
t
a
= initial amount
r
= rate, written as a percent
t
= time
Key Word
Looks like...
Formula
“increases”
Cassandra bought an antique dresser for $500. If the value
of her dresser increases 6% annually, what will be the value
of Cassandra’s dresser at the end of 3 years to the nearest
dollar
?
Exponential
Growth
“Compounded
annually”
Jack has $800 to invest. The bank offers an interest rate of
6.5% compounded annually. How much money will Jack
have after 4 years?
Exponential
Growth
“appreciates”
A painting whose initial value is $5000, appreciates at a rate
of 7.25% a year. Write a function, f
(t
),
that will model the
value of the painting after t
years.
Exponential
Growth
“depreciates”
The value of a car purchased for $20,000 depreciates at a
rate of 12% per year. What will be the value of the car after
3 years?
Exponential
Decay
“Decreased by”
Tommy started a business in the year 2001. He made a
$44,000 profit in the first year. Each year after that, his
profit decreased by 3%. Write an equation that can be used
to find P
, his profit after t
years.
Exponential
Decay
“decreases”
_____1) An initial population of 660 deer decreases at an annual rate
of 3.5%. Which exponential function models the deer
population?
Exponential
Decay
*Other keywords: “doubles,” “triples,” etc.
9
Unit 8 - Sequences
Explicit Formulas
(on Reference Sheet!)
Recursive Formulas
(Memorize format!)
Arithmetic (adding) Sequence
a
n
= a
1
+ (n
- 1)(d
)
a
1
is the 1st term in the
sequence
d
is the rate of
change/common difference
Geometric (multiplying) Sequence
a
n
= a
1
(r
)
n
- 1
a
1
is the 1st term in the
sequence
r
is the factor
Arithmetic (adding) Sequence
#a
1
=
and
+ da
n
= a
n−1
a
1
is the 1st term in the
sequence
d
is the rate of change/common
difference
Geometric (multiplying) Sequence
#a
1
=
and
ra
n
= a
n−1
a
1
is the 1st term in the
sequence
r
is the factor
10