## RELATIONS

A

**relation**can be represented in many ways: (If the x does not repeat, then the relation is a FUNCTION)## FUNCTIONS

A FUNCTION is a relation where each x goes to only one y. In other words, the x does not repeat. It will pass the vertical line test and each x has only one y.

Determining if a relation is a function:

ordered pairs: Look at the x values (the first coordinate in the ordered pair), do any of the x's repeat? If they do, it is NOT a function.

Tables: Look at the x values (left side), do any of the x values repeat? If they do, it is NOT a function.

Mappings: Look at the x values (left oval), do any of the x values have two lines - drawn to two different y values?: If they do, it is NOT a function

Graphs: Does it pass the vertical line test? If you draw a vertical line, will it cross the graph only once? Then it IS a function.

ACTIVITY: Is it a Function??

ordered pairs: Look at the x values (the first coordinate in the ordered pair), do any of the x's repeat? If they do, it is NOT a function.

Tables: Look at the x values (left side), do any of the x values repeat? If they do, it is NOT a function.

Mappings: Look at the x values (left oval), do any of the x values have two lines - drawn to two different y values?: If they do, it is NOT a function

Graphs: Does it pass the vertical line test? If you draw a vertical line, will it cross the graph only once? Then it IS a function.

ACTIVITY: Is it a Function??

## PROPERTIES

Commutative Property (order does not matter) (a)(b) = (b)(a) and a + b = b + a

Associative Property (grouping does not matter) (ab)c = a(bc) and (a + b) + c = a + (b + c)

Distributive Property a (b + c) = ab + ac

Activity: Name the Property

Associative Property (grouping does not matter) (ab)c = a(bc) and (a + b) + c = a + (b + c)

Distributive Property a (b + c) = ab + ac

Activity: Name the Property

## y = mx + b (Slope intercept Form) ...the equation for a line

- y: y is the DEPENDENT variable, the y values are the RANGE, y is also known as f(x), the 2nd coordinate in an ordered pair (__, y)

m: m is the slope, the rate of change, the change in y over the change in x, rise/run, y2-y1/x2-x1

x: x is the INDEPENDENT variable, the x values are the DOMAIN, fir first coordinte in an ordered pair (x, ___)

b: b is the y-intercept. (0, b), where the line crosses the y-axis, where x is 0, the starting point, beginning value, the constant

## SCATTERPLOTS

A SCATTERPLOT is a graph in which two sets of data are plotted on a coordinate plane

Scatterplots are used to investigate a relationship between 2 quantities and to make predictions

The independent variable belongs on the x-axis and the dependent variable belongs on the y-axis.

Scatterplots will have a positive correlation, negative correlation or no correlation

The trend line goes right through the midst of the points. It is used to make predictions.

Scatterplots are used to investigate a relationship between 2 quantities and to make predictions

The independent variable belongs on the x-axis and the dependent variable belongs on the y-axis.

Scatterplots will have a positive correlation, negative correlation or no correlation

The trend line goes right through the midst of the points. It is used to make predictions.

## Scatterplots flashcards

## DOMAIN AND RANGE

DOMAIN: the set of all possible x values

RANGE: the set of all possible y values

DoLoR the RoBoT (Domain: from Left to Right, Range: from Bottom to Top)

RANGE: the set of all possible y values

DoLoR the RoBoT (Domain: from Left to Right, Range: from Bottom to Top)

## Continuous functions:

no breaks in the graph (it includes fractional values)

it is measured

Use

Domain: -8

it is measured

Use

__<__unless its an open cirlce or infinity, then use <Domain: -8

__<__x < 6 and Range: -8 < y__<__6## Discrete functions:

the graph consists of separate points (it only includes whole number values)

it is counted

Domain: { -2, -1, 6} and Range: {3, 1, 5}

it is counted

Domain: { -2, -1, 6} and Range: {3, 1, 5}

## SOLVING FOR X

First you box the variable, cancel what's beside it, divide by what's inside the box, Reduce and you've solved it.

To Solve for x:

1) Simplify both sides (distribute and combine like terms)

2) Cancel the Little Guy (cancel the smaller x if there are x's on both side of the equal sign)

3) Box the variable and solve for x. (Box, Cancel, Divide)

ACTIVITY: Find the Like Terms

To Solve for x:

1) Simplify both sides (distribute and combine like terms)

2) Cancel the Little Guy (cancel the smaller x if there are x's on both side of the equal sign)

3) Box the variable and solve for x. (Box, Cancel, Divide)

ACTIVITY: Find the Like Terms

## SOLVING FOR Y (WRITE IN SLOPE INTERCEPT FORM)

1) BOX THE Y (include the coefficient and its sign)

2) CANCEL WHAT'S BESIDE IT (Cancel term next to box - only combine LIKE terms)

3) DIVIDE BY WHAT'S INSIDE THE BOX (Divide everywhere)

Equations need to be in slope-intercept form (solved for y) in order to identify the slope and y-intercept

OR to put into the [y=] part of the calculator

2) CANCEL WHAT'S BESIDE IT (Cancel term next to box - only combine LIKE terms)

3) DIVIDE BY WHAT'S INSIDE THE BOX (Divide everywhere)

Equations need to be in slope-intercept form (solved for y) in order to identify the slope and y-intercept

OR to put into the [y=] part of the calculator

## SLOPE - measures the steepness of a line

## KINDS OF SLOPE

Positive Slope: it goes up and to the right

Negative Slope: it goes down and to the right

Zero Slope: horizontal lines, the equations only have y (ex. y = 4)

Undefined slope: vertical lines, the equations only have x (ex. x = 4)

THE SIGN OF THE SLOPE (the sign on m) DETERMINES IF IT IS POSTIVE OR NEGATIVE)

## Steepness of Slope

The coefficient fo x (ignoring the sign) (So the ABSOLUTE VALUE OF THE SLOPE) tells is the line is steep or flat

If m is bigger than 1, it is STEEPER than the parent function (the bigger the number, the steeper the line)

If m is between 0 and 1, it is FLATTER than the parent function (the smaller the fractionm the flatter the line)

If m is bigger than 1, it is STEEPER than the parent function (the bigger the number, the steeper the line)

If m is between 0 and 1, it is FLATTER than the parent function (the smaller the fractionm the flatter the line)

## Finding Slope from a graph

Finding the slope from a graph

1. locate two points on the line

2. Count the slope (moving from one point another)

count the rise (up is positive, down is negative)

count the run (right is positive, left is negative)

3. m = rise/run (write answer as a fraction)

<-- in this example, the slope is (down 4, right 2) or -4/2 which reduces to -2.

Remember:

If the line is horizontal, the slope is 0

If the line is vertical, the slope is undefined

1. locate two points on the line

2. Count the slope (moving from one point another)

count the rise (up is positive, down is negative)

count the run (right is positive, left is negative)

3. m = rise/run (write answer as a fraction)

<-- in this example, the slope is (down 4, right 2) or -4/2 which reduces to -2.

Remember:

If the line is horizontal, the slope is 0

If the line is vertical, the slope is undefined

Video! FINDING SLOPE FROM A GRAPH

## Finding Slope from two points

Method 1: Use the Slope Formula---------------(remember the song! "Slope is, y2 minus y1....)--------------- > Label your points with x1, y1 and x2, y2

Plug values into formula and solve

Method 2:

Use the Calculator:

Use STAT. Put x values in L1 and y values in L2

STAT, Calc, 4, enter, enter

The slope will be the "a" value

## Slope from an Equation

To identify the slope in an equation, it must be in slope intercept form

1. Solve the equation for y

Box y

cancel what's beside it (only combine like terms!)

cancel by whats in box (divide everywhere!)

2. Identify slope by looking at the coefficient of x.

1. Solve the equation for y

Box y

cancel what's beside it (only combine like terms!)

cancel by whats in box (divide everywhere!)

2. Identify slope by looking at the coefficient of x.

VIDEO! Slope Intercept Form (finding m and b and graphing from an equation)

## PARALLEL AND PERPENDICULAR

__To determine if two lines are parallel or perpendicular:__Put the equations in Slope-Intercept Form (y=mx+b) (Solve for y) and compare the slopes

If the slopes are the

**same**, the lines are

**parallel**

If the slopes are

**negative reciprocals**, the lines are

**perpendicular**

__Negative Reciprocals:__The signs are opposite from each other (one is positive and one is negative)

The numbers are reciprocals of each other

## POINT SLOPE FORM

Use POINT SLOPE FORM when you are given a point and the slope.

(sometimes you need to figure out the slope!)

1. Plug in your point (put your x value in for x1 and your y value in for y1. (leave the other x and y alone)

2. Plug in your slope for m

3. Solve for y. (distribute, the box your y and cancel what's beside it - combine like terms!)

(sometimes you need to figure out the slope!)

1. Plug in your point (put your x value in for x1 and your y value in for y1. (leave the other x and y alone)

2. Plug in your slope for m

3. Solve for y. (distribute, the box your y and cancel what's beside it - combine like terms!)

Video! Point Slope Form

## X AND Y INTERCEPTS

X INTERCEPTS

x-intercepts always have y = 0. (x, 0)

x-intercepts are the solutions to the equations

Y INTERCEPTS

y-intercepts always have x = 0. (0, y)

y-intercepts are the starting points & the "b" in the equation, y = mx + b

Finding the x-intercept

Plug 0 in for y and solve for x

Finding the y-intercept

Plug 0 in for x and solve for y.

x-intercepts always have y = 0. (x, 0)

x-intercepts are the solutions to the equations

Y INTERCEPTS

y-intercepts always have x = 0. (0, y)

y-intercepts are the starting points & the "b" in the equation, y = mx + b

Finding the x-intercept

Plug 0 in for y and solve for x

Finding the y-intercept

Plug 0 in for x and solve for y.

VIDEO! Finding the x-intercept

VIDEO! Finding the y-intercept

## DIRECT VARIATION

y varies directly with x. y = kx where k is the

Recognizing Direct Variation:

The graph of direct variation is a straight, diagonal line goes through the origin.

A table, divide y/x and get the same value (k) each time, its direct variation

Solving with DIRECT VARIATION

When given an x and y: Divide y/x for the constant of variation (k)

When given 3 values (3 numbers): Solve using PROPORTION (butterfly)

__constant of variation__Recognizing Direct Variation:

The graph of direct variation is a straight, diagonal line goes through the origin.

A table, divide y/x and get the same value (k) each time, its direct variation

Solving with DIRECT VARIATION

When given an x and y: Divide y/x for the constant of variation (k)

When given 3 values (3 numbers): Solve using PROPORTION (butterfly)

## INVERSE VARIATION

y varies inversely with x. y = k/x where k is the

The graph of inverse variation is a curve that gets close, but

never touches the x axis or y axis.

A table, multiply x and y and get the same value (k) each time, its inverse

Solving with INVERSE VARIATION

When given x and y: Multiply (x)(y) for the constant of variation(k)

When given 3 values (3 numbers): solve using

INVERSE PROPORTION (set up proportion with multiplication instead of division: xy = xy.

__constant of variation__

Recognizing Inverse Variation:The graph of inverse variation is a curve that gets close, but

never touches the x axis or y axis.

A table, multiply x and y and get the same value (k) each time, its inverse

Solving with INVERSE VARIATION

When given x and y: Multiply (x)(y) for the constant of variation(k)

When given 3 values (3 numbers): solve using

INVERSE PROPORTION (set up proportion with multiplication instead of division: xy = xy.

## SYSTEMS OF EQUATIONS

a SYSTEM of equations is two or more equations working together.

The solution will be the intersection of the two lines.

A system of equation can be solved using a calculator, by graphing, by substitution and by elimination

Solving a System of Equation in the CALCULATOR

1. type both equations into [y=] (both equations must first be solved for y)

2. 2nd, trace, 5, enter, enter, enter

Writing a System from a word problem

1. Assign the variables

2, work one sentence at a time to write the information algebraically

Example: Four pickles and two nachos cost $10. (4P + 2N = 10)

If two thing are being compared - start with bigger one

Example: Hamburgers are $4 more than French Fries (h = F + 4)

The solution will be the intersection of the two lines.

A system of equation can be solved using a calculator, by graphing, by substitution and by elimination

Solving a System of Equation in the CALCULATOR

1. type both equations into [y=] (both equations must first be solved for y)

2. 2nd, trace, 5, enter, enter, enter

Writing a System from a word problem

1. Assign the variables

2, work one sentence at a time to write the information algebraically

Example: Four pickles and two nachos cost $10. (4P + 2N = 10)

If two thing are being compared - start with bigger one

Example: Hamburgers are $4 more than French Fries (h = F + 4)

VIDEO!! Writing Systems of Equations

VIDEO! Solving a System by Substitution

VIDEO! Solving a System by Graphing

VIDEO! Solving a System by Substitution

VIDEO! Solving a System by Graphing

## INEQUALITIES

1) solve the inequatlity for y (if you multiply or divide by a negative, FLIP the inequality sign)

2) draw the line using slope and y-intercept

3) the line will be dashed if it is < or >

the line will be solid if it is

4) shade above the line if the inequality is > or

shade below th eline is the inequality is < or

2) draw the line using slope and y-intercept

3) the line will be dashed if it is < or >

the line will be solid if it is

__<__or__>__4) shade above the line if the inequality is > or

__>__shade below th eline is the inequality is < or

__<__## EXPONENTS

Adding or Subtracting Exponents: no changes in exponents (its combining like terms)

Multiplying Exponents: Multiply the coefficients and ADD the EXPONENTS

Dividing Exponents: Divide tghe coefficients and SUBTRACT the EXPONENTS

Power to Power (exponent to exponent): MULTIPLY the EXPONENTS

Multiplying Exponents: Multiply the coefficients and ADD the EXPONENTS

Dividing Exponents: Divide tghe coefficients and SUBTRACT the EXPONENTS

Power to Power (exponent to exponent): MULTIPLY the EXPONENTS

## MULTIPLYING BINOMIALS

Multiplying Binomials by Box Method

1) multiply to fill in boxes

2) combine like terms (usually found on the diagonal)

<----- the answer: x^2 - 25 (the 5x and -5x cancelled)

Multipying Binomials by FOIL Method

1) multiply the frist tersm, the outer terms, the inner terms and the last terms

2) combine like terms (usually the inner and outer terms)

1) multiply to fill in boxes

2) combine like terms (usually found on the diagonal)

<----- the answer: x^2 - 25 (the 5x and -5x cancelled)

Multipying Binomials by FOIL Method

1) multiply the frist tersm, the outer terms, the inner terms and the last terms

2) combine like terms (usually the inner and outer terms)

VIDEO: MULTIPLYING BINOMIALS

## multiplying binomials practice

## FACTORING GCF

Find the largest term that will divide evenly into all terms

the GCF of 14ac - 6ab is 2a since 2a is the largest term that will divide into 14ab and 6ab evenly

Now divide each term by the GCF.

14ac– 6ab = 2a(7c - 3b)

the GCF of 14ac - 6ab is 2a since 2a is the largest term that will divide into 14ab and 6ab evenly

Now divide each term by the GCF.

14ac– 6ab = 2a(7c - 3b)

## FACTORING TRINOMIALS

1) label a, b, and c

2) put (a)(c) in the top of the X and b in the bottom

3) find the factors that will multiply to the top number and add to the bottom number

4) rewrite the equation using the two new factors for the middle term

5) factor the GCF from the first two terms (factor a negative if x is negative)

6) factor the GF from the last two terms (factor a negative if x is negative)

7) Factor the matching parenthesis (the GCF of the two remaining terms)

2) put (a)(c) in the top of the X and b in the bottom

3) find the factors that will multiply to the top number and add to the bottom number

4) rewrite the equation using the two new factors for the middle term

5) factor the GCF from the first two terms (factor a negative if x is negative)

6) factor the GF from the last two terms (factor a negative if x is negative)

7) Factor the matching parenthesis (the GCF of the two remaining terms)

## QUADRATICS

y = ax^2 + bx + c

it opens up if a is positive, it opens down if a is negative

if it opens up, the vertex is a minimum

if it opens down, the vertex is a maximum

if |a| > 1, then the parabola is narrow

if 0 < |a| < 1 (a fraction) then the parabola is wide

c is the y-intercept (0, c) (translating is sliding the parabola up or downt he y-axis)

the solutions (also called roots and zeros) are the x-intercepts

There can be two, one or no solutions

it opens up if a is positive, it opens down if a is negative

if it opens up, the vertex is a minimum

if it opens down, the vertex is a maximum

if |a| > 1, then the parabola is narrow

if 0 < |a| < 1 (a fraction) then the parabola is wide

c is the y-intercept (0, c) (translating is sliding the parabola up or downt he y-axis)

the solutions (also called roots and zeros) are the x-intercepts

There can be two, one or no solutions

## Solving Quadratics

Set equation equal to y or 0

factor , then set each factor equal to zero and solve.

OR

solve for y (or 0) and type into y=

then type 0 in y=

2nd, trace, 5, move cursor to x-intercept and enter, enter, enter (repeat for other x-intercept)

factor , then set each factor equal to zero and solve.

OR

solve for y (or 0) and type into y=

then type 0 in y=

2nd, trace, 5, move cursor to x-intercept and enter, enter, enter (repeat for other x-intercept)

## QUADRATIC FORMULA

solve the equation for y (or 0)

label a, b, and c

plug in to formula and simplify

label a, b, and c

plug in to formula and simplify

## EXPONENTIAL GROWTH AND DECAY

Exponential Growth: (grows quickly)

as x increases. y increases.

to find b, 1 + rate

Exponential Decay: (decreases quickly)

as x increases, y decreases.

to find b, 1 - rate

as x increases. y increases.

to find b, 1 + rate

Exponential Decay: (decreases quickly)

as x increases, y decreases.

to find b, 1 - rate

**Solving Quadratics by graphs, tables and concrete models**

homework

video - solve quadratics by graphs, tables and concrete models

Solving Quadratics by Quadratic Formula

homework

video - solve quadratics by quadratic formula

Form - Complete this form after viewing the video

Form - Complete this form after watching the video

homework

video - solve quadratics by quadratic formula

Form - Complete this form after viewing the video

Form - Complete this form after watching the video

**Solve Systems of Equations Using Concrete Models**

VIDEO: Solve Systems using Concrete Models (youtube)

Homework

Standard Form

VIDEO: Standard Form

Homework

VIDEO: Standard Form

Homework

Families of Lines

VIDEO: Families of Lines

Homework

VIDEO: Families of Lines

Homework

Point Slope Form

VIDEO: Point Slope Form

Homework : Day 1

Homework: Day 2

VIDEO: Point Slope Form

Homework : Day 1

Homework: Day 2

Direct Variation

VIDEO: Direct Variation

Homework

VIDEO: Direct Variation

Homework

x and y Intercepts

VIDEO: x and y Intercepts

Homework

VIDEO: x and y Intercepts

Homework

Graphing Equations in Slope-Intercept Form

VIDEO: Graphing Equations in Slope-Intercept Form

Homework

VIDEO: Graphing Equations in Slope-Intercept Form

Homework

Solve for y (Slope Intercept Form)

VIDEO: Solve for y

Homework - Day 1

Homework - Day 2

VIDEO: Solve for y

Homework - Day 1

Homework - Day 2

Slope Project

Before the Project Information

The Project

Before the Project Information

The Project

Multiple Representations and Evaluating a Function

VIDEO: Multiple Representations

VIDEO: Evaluating a Function

Homework

VIDEO: Multiple Representations

VIDEO: Evaluating a Function

Homework

First Six Weeks Test REVIEW

homework

homework

First Six Week's Skillbuiders

Skillbuilder 1 homework

Skillbuilder 2 homework

Skillbuilder 3 homework

Skillbuilder 1 homework

Skillbuilder 2 homework

Skillbuilder 3 homework

Solving Applications

homework

homework

Solving Equations with Variables on Both Sides

VIDEO: Solving Equations with Variables on Both Sides

Homework

VIDEO: Solving Equations with Variables on Both Sides

Homework

Solving Two Step Equations

VIDEO: Solving Two Step Equations

Homework

Song - Solving an Equation (Pop! Goes the Weasel)

Vocabulary

VIDEO: Solving Two Step Equations

Homework

Song - Solving an Equation (Pop! Goes the Weasel)

Vocabulary

**Combine Like Terms**

VIDEO: Combine Like Terms

Homework

Vocabulary

Find the Like Terms

Combine like terms - Practice (flashcards, matching...)

Patterns

VIDEO: Patterns

VIDEO: Patterns into a Table

VIDEO: Pattern to an Expressions

Homework

Vocabulary

VIDEO: Patterns

VIDEO: Patterns into a Table

VIDEO: Pattern to an Expressions

Homework

Vocabulary

Variables and Evaluating Expressions

VIDEO: Variables and Evaluating Expressions

Homework

Vocabulary

Phrase Sort

VIDEO: Variables and Evaluating Expressions

Homework

Vocabulary

Phrase Sort

Scatterplots

VIDEO: Scatterplots

Homework

Vocabulary

VIDEO: Scatterplots

Homework

Vocabulary

Integers
VIDEO - Integers Homework |