Algebra 1 Regents Exam Study Guide New

Algebra Regents Exam 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

Multiple Choice Strategies Test Taking Strategies

● Compare your Choices Free Response Strategies
● Eliminate Choices
● Guess and Check ● Read carefully
● Multiple Methods of Solution ● Highlight keywords
● Process of Elimination ● Break the question into chunks that you understand.
● Use your Calculator ● 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 d​ oes not​ lie on the line”
“No, because it d​ oes 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? “Yes, because it lies in the double shaded region”
(system of inequalities)
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 “Initial amount of _______”
(keywords, see page 9) “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”
Determine and state all values of x​ ​ for which f​ ​(x​ ​) = ​g​(​x​). Where the two graphs intersect.

State whether the vertex represents a m​ aximum​ point. The leading coefficient is p​ ositive​.

State whether the vertex represents a m​ inimum​ point. The leading coefficient is n​ egative.​

Assess the fit of the line for residuals. Is a good fit because there is not a pattern.

Not a good fit because there is a pattern.

Strong p​ ositive​ correlation. Correlation coefficient is very close to 1.

Strong n​ egative c​ orrelation. Correlation coefficient very close to -1.

**(In the context of the problem)** As ____________ increases, _____________ also increases.
Strong p​ ositive​ correlation. As ____________ increases, _____________ decreases.

Strong ​negative c​ orrelation.

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Unit 1 - Polynomials

Adding/Subtracting Polynomials Multiplying Polynomials/Distributive Property
(Box Method)

1. Identify terms that have the same base and same 1. Multiply coefficients.
exponent. 2. Keep the base.
3. Add the exponents.
2. Add coefficients. 4. Combine like terms (see box to the left).
3. Keep the base and the exponent.
4. Standard form - greatest exponent first and then

follow in descending order.

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”

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Unit 2 - Equations

Example of Solving an Equation Example of Solving a Literal Equation

Given the formula d = 1 at2 , solve for ​t.
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” “x​ ​is greater than 4” “​x​ is less than or equal to 4” “​x ​is greater than or equal to 4”
Open circle Open circle Closed circle Closed Circle

Shade left Shade right Shade left Shade right

Interval Notation Meaning Compound Inequality Graph

(-2, 3) -2 is not included -2 < ​x​ < 3
3 is not included
(-2, 3] -2 < ​x​ ≤ 3
{-1, 0, 1, 2}
[-2, 3) -2 ≤ ​x​ < 3
-2 is not included
[-2, 3] 3 is included -2 ≤ x​ ​ ≤ 3
{-1, 0, 1, 2, 3}

-2 is included
3 is not included
{-2, -1, 0, 1, 2}

-2 is included
3 is included
{-2, -1, 0, 1, 2, 3}

≤ ≥

“at most” “at least”
“no more than” “no less than”

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Unit 4 - Linear Equations and Inequalities

Graphing ​Line​ar 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. In order for a point to be a solution it must be on the
If true, then it is a solution. table and line.

If false, then it is not a solution.

Rate of Change “Rise over Run”

Graphing Inequalities Example

Notes

● <​ 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!

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Unit 5 - Systems of Linear Equations & Inequalities

Linear System Inequality System

● Graph each line separately! ● Graph each inequality separately!
● The point of intersection is your answer. This ○ <​ ​ is a dashed line and shade down.
○ >​ is a dashed line and shade up.
is called your ​“Solution.” ○ ≤ is a solid line and shade down.
● Remember - you must have tables, arrows,
○ ≥ is a solid line and shade up.
and labels for everything in order to get full ● The double shaded section is your ​“Solution.”
credit on your midterm!
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 *Make sure equations are in “​y”​ = format.

negative (multiply top on bottom, and bottom on top).

● Step 2: Distribute to each term. ● Step 1: Set equations equal to each other.

● Step 3: Combine like terms and solve for y​ ​. ● Step 2: Solve for ​x.​

● Step 4: Substitute ​y​ into one equation and solve for x​ ​. ● Step 3: Substitute x​ ​ into one equation and solve for y​ ​.

● Your solution should be written as a point (x​ ​, ​y)​ ● Your solution should be written as a point (​x,​ ​y​) unless

unless it is a real world situation. it is a real world situation.

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Key Feature Unit 6 - Functions Example
Domain
Range Meaning
“Input” of a function (x​ ​-values).
“Output” of a function (​y​-values).

Domain: A​ ll Real Numbers
Range:​ f (x) ≥ 2

*Each input can only have 1 output.*

(see page 2 for free - response catchphrases)

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Unit 7 - Exponential Growth & Exponential Decay

Exponential Growth Formula Exponential Decay Formula
(Memorize!) (Memorize!)

f (t) = a(1 + r)t f (t) = a(1 − r)t

a​ = initial amount a​ = initial amount
r​ = rate, written as a percent r​ = rate, written as a percent

t​ = time t​ = time

Key Word Looks like... Formula

“increases” Cassandra bought an antique dresser for $500. If the value Exponential
Growth
of her dresser i​ ncreases​ 6% annually, what will be the value
of Cassandra’s dresser at the end of 3 years to the n​ earest
dollar​?

“Compounded Jack has $800 to invest. The bank offers an interest rate of Exponential
annually” 6.5% c​ ompounded annually.​ How much money will Jack Growth
have after 4 years?

“appreciates” A painting whose initial value is $5000, ​appreciates​ at a rate Exponential
of 7.25% a year. Write a function, ​f(​ t​ ​),​ t​ hat will model the Growth
value of the painting after t​ ​years.

“depreciates” The value of a car purchased for $20,000 ​depreciates​ at a Exponential
rate of 12% per year. What will be the value of the car after Decay

3 years?

“Decreased by” Tommy started a business in the year 2001. He made a Exponential
Decay
$44,000 profit in the first year. Each year after that, his
profit d​ ecreased by​ 3%. Write an equation that can be used
to find ​P​, his profit after ​t​ years.

“decreases” A​ n initial population of 660 deer d​ ecreases​ at an annual rate Exponential
of 3.5%. Which exponential function models the deer Decay
population?

*Other keywords: “doubles,” “triples,” etc.

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Unit 8 - Sequences

Explicit Formulas Recursive Formulas
(on Reference Sheet!) (Memorize format!)

● Arithmetic (adding) Sequence ● Arithmetic (adding) Sequence

○ a​n =​ a​ ​1​ ​ + (​n​ - 1)(​d)​ ○ a1 = #​
■ a​1 ​is the 1st term in the and
sequence
■ d​ is the rate of an = an−1 + d​
change/common difference ■ a​1 ​is the 1st term in the
sequence
● Geometric (multiplying) Sequence ■ d​ is the rate of change/common
■ difference
○ a​n ​= ​a1​ ​(r​ ​) n​ ​ - 1
■ a1​ ​is the 1st term in the ● Geometric (multiplying) Sequence a1 = ​#
sequence and
■ r​ is the factor
an = an−1• r
■ a​1 ​is the 1st term in the
sequence
■ r​ is the factor

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Unit 9 - Quadratics Part 1

Greatest Common Factor – “GCF” 1. Factor out the LARGEST number and/or variable.
2. Divide the expression by the GCF.
Sum/Product – 3. Place the GCF on the outside of the parenthesis and the new
(trinomials only!)
expression on the inside of the parenthesis.
ax2​ ​ + ​bx​ + ​c
1. Write the equation in standard form.
Difference of Perfect Squares – “D.O.P. S” 2. Look for a GCF.
(binomials only!) 3. Find factors whose sum is “b” and whose product is “c.”
4. Factor!
5. Quick check!

1. Take the square root of each term.
2. Separate answers into two parentheses.
3. Put an addition sign in one parenthesis and a subtraction sign

in the other.

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Unit 10 - Quadratics Part 2

Vocabulary

Systems Completing the Square

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Unit 11 - Solving Quadratic Equations

Perfect Square List Solve by Completing the Square

√1 = 1 Ex: x2 + 8x + 4 = 0
√4 = 2
√9 = 3 1. Bring the “c” value to the other side and
√16 = 4 add blanks
√25 = 5 x2 + 8x +−− = − 4 +−−
√36 = 6
√49 = 7 2. Divide the “b” value by 2 and square it
√64 = 8 (x + 4)2 = 12
√81 = 9
√100 = 100 3. Take the square root of both sides
√121 = 11
√(x + 4)2 = ± √12

4. Simplify
x = − 4 ± √12
x = − 4 ± 2√3

How many real roots are there? Quadratic Formula

1. 1 Real Solution​ - you will get the (given to you on the reference sheet)
same number​ for each root.
(the graph will cross the x-axis x= −b±√b2−4ac
ONCE)
2a

2. 2 Real Solutions​ - you will get 2​ ** Make sure the equation is =0!**
different numbers ​for each root. ** MUST substitute with parentheses!**
(the graph will cross the x-axis
TWICE) Ex: x2 − 2x − 6 = 0
a= 1, b= -2, c= -6
3. No Real Solutions​ - you will get a
negative number​ under the √ x = −(−2)±√(−2)2−4(1)(−6)
symbol.
(the graph will NEVER cross the 2(1)
x-axis)

13

x = 2±√28
2

Solving Quadratics by Factoring Factoring Reminders

Ex: x2 + 2x − 8 = 0 1. Sum and Product (Add/Multiply)
1. Make sure the equation = 0 and
x2 − 8x + 15
factor the quadratic. = (x − 3)(x − 5)

(x + 4)(x − 2) = 0

2. Set each factor =0 by making a 2. DOPS
“T-chart” and solve for “x”.
x2 − 9
(x + 4)(x − 2) = 0 = (x + 3)(x − 3)

x+4=0 x−2=0

x= −4 x=2 3. GCF

2x2 + 16x
= 2x(x + 8)

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Unit 12 - Functions Part 2

Translations Scaling and Reflecting
1. INSIDE​ the parent function:
1. If coefficient is:
2. OUTSIDE​ the parent function: a. GREATER than 1:
i. The graph gets
NARROWER
b. Between 0 and 1:
i. The graph gets
WIDER

2. If coefficient is NEGATIVE:
a. The graph gets
REFLECTED over the
x-axis or opens
DOWNWARD

Piecewise Functions Systems of Equations

★ < or > = open circle ❖ Make sure ALL equations are in
y = mx + b form.
★ ≤ or ≥ = closed circle
❖ REMEMBER:
★ ONLY graph the points given to you ➢ Absolute value is a
in the domain! V-shape
○ If the domain looks like ➢ Quadratic is a U-shape
# ≤ x < # , you DO NOT put
arrows on the end ❖ The graphs WILL cross each
○ If the domain looks like either other at one OR more than one
x < # or x > # , you DO put point.
arrows on the end

15

Unit 13 - Statistics

Calculator Steps for Calculator Steps for
1 Variable Statistics Linear & Exponential Regression

16

Correlation and Causation

Box Plots

Dot Plots Example of Symmetric Example of Skewed

Scatter Plots ● Plot data but do NOT connect the dots!
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Residuals

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