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Algebra 1

Handbook and Notes

Student Name: Class: Year:

Algebra 1—Handbook and Notes

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Class Goals Our goals for the Algebra 1 course include:

1. Successfully preparing for and passing the End of Instruction Exam 2. Successfully preparing for the next level of math:

a. Geometry b. Algebra 2 c. Precalculus d. Math Finance

3. Successfully preparing for the ACT Exam 4. Learn how to apply math skills and concepts to real-world situations

Student Work Goals Doing quality work is extremely important in both the academic world (i.e., school) and the real-world (i.e., home and work). Therefore, everything we do in class should include these characteristics:

1. Neatness—our work is organized and easy to read 2. Detail—our work is clear in meaning and not vague 3. Accuracy—our work is correct and free from errors

Desired Employee Skills The following three skills were identified in the book by Bill Coplin entitled 10 Things Employers Want You to Learn in College. Below are three of those skills employers will be looking at as they consider hiring you for a job opening. These same skills will also help you as you continue to pursue your education in high school. Therefore, high school is a great time to begin practicing these skills.

1. Work well with others a. Be kind, friendly, and patient with others b. Be a team player

2. Know your numbers a. Working with data is important in every work field

3. Be responsible for yourself a. Have strong motivation b. Be ethical (i.e., do the honest and right thing) c. Have good time-management skills (turn projects/assignments in by their

deadline)


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Class Notes Example The importance of taking class notes in a math course:

1. Allows the student to practice doing the problems with the instructor 2. Creates a resource of information and examples for the student to refer back to when

doing assignments and studying 3. Allows the teacher to help the student when he or she has extra questions 4. Shows that the student is putting effort into their own learning.

The example below shows what high quality notes should look like. Notice, they also meet our Student Work Goals in that they are neat, detailed, and accurate.

3.3 Solving Linear Equations

Sherry Walker 4th Hour 10/3/14

3.3

Solving Linear Equations:

a. Get 𝒙 by itself on one side of the equal sign

b. Always do the opposite of what is being done to both sides of the

equation

c. Work in reverse order of operations or think of it as “do what is with

the variable last”.

Example 1:

Solve for 𝑥.

3𝑥 + 4 = 16

−4 −4

3𝑥 = 12

3 3

𝑥 = 4

Example 2:

A plumber charges $35 for a service call fee and $8 per hour for repairs. Write and

solve an equation for how many hours the plumber worked if he charges a

total of $83.

83=8x+35

-35 -35

48=8x

8 8

X=6

The plumber worked a total of six hours when charging $8 per hour and

an initial $35 service call fee.


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Homework Assignment Example Homework guidelines:

1. You must include your complete heading. 2. You must show detailed work when solving problems. 3. Application problems (i.e., word problems) require the answer to be written as a

complete sentence with a subject, verb, solution, and proper punctuation.

Sherry Walker

4th Hour

10/3/14

3.3

1. 1

3 𝑥 − 6 = −8

+6 +6

1

3 𝑥 = −8

(3)1

3 𝑥 = −8(3)

𝑥 = −24

2. 5𝑥 + 3(𝑥 + 4) = 28

5𝑥 + 3𝑥 + 12 = 28

8𝑥 + 12 = 28

−12 −12

8𝑥 = 16

8 8

𝑥 = 2

3. 𝑐 = 12(𝑥) + 45

𝑐 = 12(4) + 45

𝑐 = 48 + 45

𝑐 = 93

The electrician would charge a total of $93 for working four hours at $12 an hour

and charging a $45 service fee.

4.


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Resources Teacher Ease Teacher Ease is the website you will log into to access assignments, videos, tutorials, and grades. Teacher Ease can be accessed at the following address:

http://www.teacherease.com Holt Online Textbook and Assignments You can access your entire textbook, worksheets, tutorial videos directly on the textbook company’s website. Log on below with the login and password you have been provided at the address below:

http://my.hrw.com Chromebook Apps It is recommended that the following Google Chrome applications be installed on your Chromebook:

1. General Apps a. Google Drive b. Google Docs c. Google Sheets d. Google Calendar e. Google Search

2. Math Apps a. Desmos Graphing Calculator b. TeX equation editor c. Daum Equation Editor (As a backup to TeX equation editor) d. GeoGebra

Calendars

1. District Calendar Link

2. High School Events Calendar Link

3. Algebra 1 Calendar Link http://www.google.com/calendar/feeds/cushing.k12.ok.us_sfl6jo41h8jhllds28p5dq1k2s%40group.calendar.google.com/public/basic

http://www.teacherease.com/

http://my.hrw.com/

http://www.google.com/calendar/feeds/cushing.k12.ok.us_sfl6jo41h8jhllds28p5dq1k2s%40group.calendar.google.com/public/basic

http://www.google.com/calendar/feeds/cushing.k12.ok.us_sfl6jo41h8jhllds28p5dq1k2s%40group.calendar.google.com/public/basic


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End of Instruction (EOI) Exam Examples The End of Instruction (EOI) Exam will be made up of not only multiple choice questions, but also constructed response questions that will require students to:

justify their steps in solving a problem

show their work

explain their steps in solving a problem

explain the meaning of the answer using complete sentences

sketch functions by hand Examples of each of these types of questions are shown below. Multiple Choice

1. Consider this quadratic equation. 𝑥2 + 2𝑥 − 48 = 0

What are the solutions to the equation? (A.REI.4b)

A 6 and −8

B 8 and −6

C 4 and −12

D 12 and −4

Constructed Response

2. A generator contains 20 gallons of gasoline. It consumes the gasoline at a rate of 0.9 gallons per hour. The generator fuel light turns on when the gasoline level drops to 2 gallons. The equation shown can be used to determine the number of hours, h, the generator will run before the fuel light turns on.

20 − 0.9ℎ = 2 Solve the equation for ℎ to determine the number of hours before the fuel light turns on. Justify why each step was used to solve the equation. (A.REI.1)

Notice


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3. Solve the following system of equations. Find the ordered pairs that satisfy both

equations. (A.REI.11) 2

3𝑥 +

3

5𝑦 = 3

1

2𝑥 −

3

5𝑦 = 4

Show your work.

4. Miranda is training for a 26-kilometer race to raise money for her favorite charity. The first week of training, she runs 6 kilometers. She then adds 2 kilometers to her training run each week.

She uses the equation t = 2w + 4, where t represents the total kilometers ran and w represents the number of weeks.

Solve the equation for w and explain the operation used in each step. (A.REI.1)

In which week of her training program will she reach 26 kilometers?

Notice

Notice Notice


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5. The owner of a new theme park needs to determine what ticket price will result in the largest attendance and therefore the maximum revenue. The predicted daily revenue for a large theme park is given by the function 𝑓(𝑥) = 40,000𝑥 − 250𝑥2, where x is the admission price in dollars per person. Find the 𝑥-intercept(s) and the vertex of the graph of 𝑓(𝑥). (A.IF.4) Show your work. 𝑥-intercepts

Vertex Explain the meaning of the x-intercept(s) and the vertex in this situation.

Sketch a graph of 𝑓(𝑥) on this coordinate plane.

Notice

Notice

Notice


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Introduction Unit Order of Operations

1. Grouping Symbols—parentheses: ( ), absolute value: | |, radicals: √ 2. Powers—exponents 3. Multiplication/Division left to right 4. Addition/Subtraction left to right

Absolute Value—the distance that a number is from zero (the origin) on the real number line Ex. |3| = 3 Ex. |−3| = 3 Algebraic Expression

A collection of numbers, variables, and operations

Coefficient Exponent 3𝑥2 + 5

Variable Operation Constant

Coefficient—the number in front of a variable Exponent—a number written as a superscript that represents the number of times the base is

used as a factor Variable—a letter used to represent one or more numbers Operation—addition, subtraction, multiplication, division Constant—a plain number with no variable

Distance Formula 𝑑 = 𝑟𝑡 𝑑 = distance 𝑟 = rate 𝑡 = time

Simple Interest Formula 𝐼 = 𝑝𝑟𝑡 𝐼 = simple interest 𝑝 = principal (initial amount) 𝑟 = interest rate 𝑡 = time in years


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Expressions, Equations, and Inequalities Expression—a collection of numbers, variables, and or operations Ex. 3𝑥 + 𝑦 Equations—has an equal sign (=) between two expressions Ex. 2𝑥 + 4 = 19𝑥 − 7 Inequality—has an inequality symbol (<, >, ≤, ≥) between two expresions Ex. 16𝑚 ≥ 96

Translating Verbal Phrases

“is” (also verbs such as “are” and “will be”) —means =

“more than” and “and”—usually mean addition

“less than” and “difference”—mean subtraction

“of”—usually means multiplication

“3 feet per second”—usually can be written as 3𝑥


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Real Numbers

real number—all rational and irrational numbers Examples: every number that exists

rational number—a number 𝑎

𝑏 where 𝑎 and 𝑏 are intergers and 𝑏 ≠ 0

Examples: 2, −.45, −2

3

irrational number—a number that cannot be written as a quotient of two integers (i.e., a number that cannot not be written as a fraction

Examples: √17, 𝜋, −√3 Square Roots

radical symbol

√𝑎 radicand

square root—if 𝑏2 = 𝑎, then 𝑏 is a square root of 𝑎. Example: 32 = 9 and (−3)2 = 9, so 3 and −3 are square roots of 9. perfect squares—the square of an integer

Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225


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Solving Linear Equations Tips for Solving Linear Equations:

Get 𝑥 by itself on one side of the equal sign.

Always do the opposite operation that is being done and do it to both sides of the equation.

Work in reverse order of operations. (Another way to think of this is to “do what’s with the variable last.”)

Word Problem Key Words “is” (also verbs such as “are” and “will be”) —means = “more than” and “and”—usually mean addition “less than” and “difference”—mean subtraction “of”—usually means multiplication “3 feet per second”—usually can be written as 3𝑥 Plumber Problem Example A local plumber charges customers a $35 service call fee for diagnosing a plumbing problem and $8 per hour for labor. Write an equation for the total cost of the plumber’s charges.

Total Cost Number of Initial Service Hours Worked Call Fee

𝑐 = 8𝑥 + 35

“is” (verb) “$8 per hour” Rates, Ratios, and Proportions Rate—compares different units by division

Examples: miles per hour miles

hour cost per person

cost

person

1 yard = 3 feet can be written as:

1 yard

3 feet or

3 feet

1 yard

Ratio—compares same units by division


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Functions Function – a rule that establishes a relationship between two quantities.

“for every input, there is exactly one output” ← KEY

input – what you plug into an equation output – what comes out of an equation (essentially, your answer) domain – all of the input values (what you can plug into the equation) range – all of the output values (think of it as your “range” of answers)

Domain Inputs

𝑥 Independent variable

Range

Outputs 𝑦

Dependent variable

Function Notation 𝑓(𝑥) = 𝑦 Example: 𝑦 = 3𝑥 + 5 𝑓(𝑥) = 3𝑥 + 5 Function notation Types of Functions (In Algebra 1)

1. Linear Function—a line; increases by a constant rate of change (same slope)

𝑓(𝑥) = −4𝑥 + 7

2. Absolute Value Function—a “v” shape; can open up or down

𝑓(𝑥) = 2|𝑥 + 3| − 6

3. Piece-wise Function—made of different “pieces”, each with a different domain

f(x) = {x − 3, if x ≥ 0

−5x + 2, if x ≤ 0

4. Exponential Function—an exponential curve; increases or decreases by a multiply (i.e.,

percentage)

𝑓(𝑥) = 3 ∙ 5𝑥

5. Quadratic Function—a parabola; can open up or down

𝑓(𝑥) = 4𝑥2 + 7𝑥 − 1

or

or

or


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Linear Functions Linear Function

The 𝑥 and 𝑦 of a linear equation are: a. To the first power b. Not multiplied together c. Not in the denominator

Has a graph that forms a straight line

Has the same slope (i.e., rate of change) between each point on the line

Can be written in any of the three following forms: Forms of Linear Equations

Standard Form 𝐴𝑥 + 𝐵𝑦 = 𝐶

Slope-Intercept Form 𝑦 = 𝑚𝑥 + 𝑏

Point-Slope Form 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

Equations of Horizontal and Vertical Lines

The graph of 𝑦 = 𝑏 is a horizontal line. The line passes through the point (0, 𝑏).

The graph of 𝑥 = 𝑎 is a vertical line. The line passes through the point (𝑎, 0)

Parent Graph of a Linear Function

𝑦 = 𝑥

𝑥 𝑦 −2 −2 −1 −1 0 0 1 1 2 2


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Slope Formula

Slope = rise

run

𝑚 =𝑦2 − 𝑦1

𝑥2 − 𝑥1

Classification of Lines by Slope

A line with positive slope rises from left to right.

A line with negative slope falls from left to right.

A line with zero slope is horizontal.

A line with undefined slope is vertical.

𝑚 > 0 𝑚 < 0 𝑚 = 0 𝑚 is undefined

Methods of Graphing Linear Functions

1. Table a. Make and input-output table with at least five values for 𝑥. b. Plot the ordered pairs on a coordinate plane. c. Draw a line through the points with a ruler.

2. Find the 𝑥-intercept and the 𝑦-intercept a. Plug in zero to 𝑥 and solve for 𝑦 to get the 𝑦-intercept. b. Plug in zero to 𝑦 and solve for 𝑥 to get the 𝑥-intercept. c. Plot the two intercepts on a coordinate plane. d. Draw a line through the points with a ruler.

3. Slope-Intercept Form a. Rewrite the equation in slope-intercept form: 𝑦 = 𝑚𝑥 + 𝑏. b. Plot the 𝑦-intercept using the 𝑏-value. c. Plot several other points of the line using the slope (𝑚) to count rise over run

from the y-intercept. d. Draw a line through the points with a ruler.


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Slope-Intercept Form slope 𝑦-intercept

𝑦 = 𝑚𝑥 + 𝑏

rate of change original amount

Parallel and Perpendicular Lines

Parallel Lines

1. Never intersect 2. Have the same slope

a. 𝑦 = 2𝑥 + 7 b. 𝑦 = 2𝑥 − 3

Perpendicular Lines

1. Intersect at a right angle (i.e., 90o) 2. Have the opposite reciprocal slope;

Their product is −1 a. 𝑦 = 3𝑥 + 9

b. 𝑦 = −1

3𝑥 + 2


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Writing Linear Equations Forms of Linear Equations

1. Standard Form 𝐴𝑥 + 𝐵𝑦 = 𝐶

2. Slope-Intercept Form 𝑦 = 𝑚𝑥 + 𝑏

3. Point-Slope Form 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

You need two things to write the equation of a line:

1. slope 2. 𝑦-intercept

Writing the equation of a line:

1. Find the slope (𝑚) using the slope formula. 2. Plug the slope and an ordered pair that is on the line into 𝑦 = 𝑚𝑥 + 𝑏 and solve for 𝑏. 3. Write the equation of the line by plugging in the values of 𝑚 and 𝑏 to 𝑦 = 𝑚𝑥 + 𝑏.

Create a Line of Best Fit

1. Make a scatter plot of the data. 2. Draw a line of best fit. 3. Pick two points on the line and find the slope of the line using the slope formula. 4. Plug the slope and an ordered pair that is on the line into 𝑦 = 𝑚𝑥 + 𝑏 and solve for 𝑏.


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Linear Inequalities Inequality Symbols

< Less than

> Greater than

≤ Less than or equal to

≥ Greater than or equal to Graphing Linear Inequalities

1. Graph the inequality as you would an equation a. Used a dashed line for < and > b. Used a solid line for ≤ and ≥

2. Shade the side of the line with all the solutions to the inequality. a. Option 1: Test a point on either side of the line to determine if the point is a solution. If

the point is a solution, shad that side of the line. b. Option 2: If the inequality is in slope-intercept, shade according to the inequality

symbol. i. Example 1: For 𝑦 < 2𝑥 + 3, shade below the line because that is where “𝑦 is

less than” than the line. ii. Example 2: For 𝑦 > 2𝑥 + 3, shade above the line because that is where “𝑦 is

greater than” the line.

Graphing Linear Inequalities

Shaded Below “less than or equal to 𝑦”

𝑦 ≤ 2𝑥 + 3

Shaded Above “greater than y”

𝑦 > 2𝑥 + 3


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Systems of Linear Equations Methods for Solving Linear Systems

1. Graphing a. Write each equation in slope-intercept form: 𝑦 = 𝑚𝑥 + 𝑏. b. Graph both equations on the same coordinate plane. c. The point (i.e., ordered pair) where the lines intersect is the solution to the linear

system. d. Write your answer as an ordered pair.

2. Substitution a. Solve one of the equations for one of its variables. When possible, solve for a

variable that has a coefficient of 1 or −1. b. Substitute the expression from Step A into the other equation and solve for the

other variable. c. Substitute the value from Step B into the revised equation from Step A and

solve. d. Write your answer as an ordered pair.

3. Elimination/Combination a. Arrange the equations with like terms in columns (such as Standard Form). b. Multiply one or both of the equations by a number to obtain coefficients that are

opposites for one of the variables. c. Add the equations from Step B. Combining like terms will eliminate one variable.

Solve for the remaining variable. d. Substitute the value obtained in Step C into either of the original equations and

solve for the other variable. e. Write your answer as an ordered pair.

Number of Solutions of a Linear System

Lines Intersect One solution 𝑥 = 𝑎 and 𝑦 = 𝑏

Lines are parallel No solution “False Statement”

Lines coincide Infinitely man solutions “True Statement”


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Exponents Rules of Exponents

1. Product of Powers—to multiply powers with the same base, add the exponents

𝑥2 ∙ 𝑥5 = 𝑥7

2. Power of a Power—to find a power of a power, multiply the exponents.

(𝑦3)4 = 𝑦12

3. Power of a Product—to find a power of a product, multiply the exponents of all the factors

(𝑥2𝑦𝑧4)5 = 𝑥10𝑦5𝑧20

4. Quotient of powers—to divide powers with the same base, subtract the exponents

𝑚5

𝑚3= 𝑚2

5. Power of a Quotient—to find the power of a quotient, multiply the exponents of all the factors

(𝑎4

𝑏3)

2

=𝑎8

𝑏6

6. Zero Exponent—all nonzero numbers to the zero power is 1

50 = 1

7. Negative Exponent—take the reciprocal and switch the sign of the exponent

𝑎−3 =1

𝑎3 or

1

𝑎−3= 𝑎3


22

Adding, Subtracting, & Multiplying Polynomials Polynomial—a monomial or the sum of monomials

1. Monomial—a number, variable, or a product of a number and one or more variables Examples: 7, 3𝑥, and 5𝑚3

2. Binomial—the sum of two monomials Examples: 2𝑥 + 4 and 9𝑦3 − 1

3. Trinomial—the sum of three monomials Example: −2𝑥2 − 3𝑥 + 6 Standard form of a polynomial:

Leading coefficient Degree Constant term

2𝑥3 + 5𝑥2 − 4𝑥 + 7


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Factoring Polynomials Factoring Cloud

1. Pull out the Greatest Common Factor (GCF) if possible. 2. Continue factoring by:

a. Difference of two squares—two terms b. Trinomial (British Method)—three terms c. Grouping—four terms

3. Check to see if it will factor again.


24

Absolute Value Functions Absolute Value Function

Has a graph that looks like a “v”

Parent Graph of an Absolute Value Function

𝑦 = |𝑥|

𝑥 𝑦 −2 2 −1 1 0 0 1 1 2 2

Is “like” the slope of Opens up if + Shifts the graph up or down. The right side of the graph Opens down if –

𝑦 = 𝑎|𝑥 − ℎ| + 𝑘 Shrinks if |𝑎| > 1 Shifts the graph left or right Stretches if 0 < |𝑎| < 1 (opposite of what you see).

(ℎ, 𝑘) is the vertex.


25

Quadratic Functions

Parent Graph of a Quadratic Function

𝑦 = 𝑥2

𝑥 𝑦 −2 4 −1 1 0 0 1 1 2 4

Forms of Quadratic Functions

1. Standard Form 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐

2. Vertex Form 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘

3. Intercept Form 𝑦 = 𝑎(𝑥 − 𝑝)(𝑥 − 𝑞)

Graphing Standard Form 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄

The x-value of the vertex is 𝑥 = −𝑏

2𝑎

The axis of symmetry is the vertical line 𝑥 = −𝑏

2𝑎

Steps for graphing:

1. Find the vertex using 𝑥 = −𝑏

2𝑎

2. Make a table with points on either the left or right of the vertex and make a table. Graphing Vertex Form 𝒚 = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌

The vertex is (ℎ, 𝑘)

The axis of symmetry is 𝑥 = ℎ Steps for graphing:

1. Find the vertex using (ℎ, 𝑘). 2. Make a table with points on either the left or right of the vertex and make a table.


26

Graphing Intercept Form

The 𝑥-intercepts are 𝑝 and 𝑞.

The axis of symmetry is halfway between (𝑝, 0), and (𝑞, 0). Steps for graphing:

1. Plot the 𝑥-intercepts using (𝑝, 0) and (𝑞, 0). 2. Find the axis of symmetry and vertex halfway between the 𝑥-intercepts. 3. Plug in 𝑥-value of the axis of symmetry into the function to find the 𝑦-value of the

vertex. Methods for Solving Quadratic Equations

1. Graphing a. Set the equation equal to zero. b. Write and graph the related function c. Identify the 𝑥-intercepts (i.e., zeros)

2. Factoring a. Set the equation equal to zero. b. Factor. c. Set each factors equal to zero and solve for 𝑥.

3. Square Root Method a. Isolate the 𝑥2 or (𝑥 − ℎ)2 b. Take the square root of both sides

4. Completing the Square

a. Create a perfect square trinomial by finding the 𝑐-value using (𝑏

2)

2

.

b. Factor the perfect square trinomial. c. Solve for 𝑥 by using the square root method.

5. Quadratic Formula

𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎

6. Synthetic Division (Algebra 2 concept)


27

Exponential Functions Exponential Function

Has a graph that is curved

The graph increases or decreases by a multiple (percentage) Exponential Growth

𝑦 = 𝑎𝑏𝑥 𝑎 is the initial amount. 𝑟 is the growth rate.

𝑦 = 𝑎(1 + 𝑟)𝑡

1 + 𝑟 is the growth factor. 𝑡 is the time period.

𝑎 > 0 and 𝑏 > 1

Exponential Decay

𝑦 = 𝑎𝑏𝑥 𝑎 is the initial amount. 𝑟 is the decay rate.

𝑦 = 𝑎(1 − 𝑟)𝑡

1 + 𝑟 is the decay factor. 𝑡 is the time period.

𝑎 > 0 and 0 < 𝑏 < 1

{

{


28

Piece-wise Functions Graphing a Piece-wise Function:

1. Make a table for each “piece” of the function. 2. Plot the points for each piece on the same coordinate plane. 3. Use and appropriately at the end(s) of each piece.

Piecewise Function

𝑓(𝑥) =2𝑥,

𝑥 − 4, 𝑖𝑓 𝑥 > 1𝑖𝑓 𝑥 ≤ 1

𝑓(𝑥) = 2𝑥 𝑓(𝑥) = 𝑥 − 4 x y x y

1 2 1 −3

0 0 2 −2

−1 −2 3 −1

−2 −4 4 0

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