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

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“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).

Rational vs. Irrational Numbers

Example of Substitution

● Rational Number - all real numbers that are not

irrational.

○ Example(s): 5, 3.5, 4.23232323...

● Irrational Number - non-terminating,

non-repeating decimal.

○ Example(s): Pi

*See the page 2 for catch phrases for extended response

questions.

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

)

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● 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

r•a

n

= a

n−1

■ a

1

is the 1st term in the

sequence

■ r

is the factor

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