w elcome to c hapter 4!. psae p ractice 1. you are the new maintenance person for the local school...

WELCOME TO CHAPTER 4!

PSAE PRACTICE1. You are the new maintenance person for the local school and need to calculate how much wax will be required for the cafeteria floor based on its square footage. The cafeteria is 120 feet by 45 feet. What is the square footage you should use to figure the amount of wax needed for the cafeteria floor?a) 165 ft2 b) 100 ft2 c) 1000 ft2 d) 5400 ft2 e)

6000 ft2

2. You must set up tables for a wedding reception in the restaurant where you work. There will be 24 individuals and 40 couples attending. Each table seats 8 persons. How many tables should you set up?a) 3 b) 5 c) 10 d) 12 e) 13

4.1 – COORDINATESOBJECTIVE: TO PLOT POINTS AND NAME

POINTS IN THE COORDINATE PLANE.

A is formed by two real number lines that intersect at the origin. (x-axis and y-axis)

An is a point in the coordinate plane represented by real numbers. The x-coordinate is the first number. The y-coordinate is the second number.

Ex. (3,6)(x, y) (right or left, up or down)

THINK OF COORDINATE POINTS AS AN…

(x, y) You must move (horizontally) into the elevator before you can go up or down.

COORDINATE PLANE (X, Y)

x- axis

y-axis

Origin

(0,0)

Quadrant I

(+, +)

Quadrant II

(-, +)

Quadrant III

(-, -)

Quadrant IV

(+, -)

PLOTTING POINTS

Then 4 up (positive) make a point

Plot these points:

1. (-2, -4)

2. (0, 3)

3. (-1,0)

4. (6,-2)

5. (-4, 5)

To plot a point: (3,4) Start at (0,0) Move 3 to the right (positive)

PRACTICE

Name the following points and give the quadrant or axis where they lie.

A:

B:

C:

D:

A

B

C

D

B

C

DE

F

Identify the ordered pairs on the coordinate plane. Name the quadrant it is in or the axis it is on.

A ________ Quad _____

B ________ Quad _____

C ________ Quad _____

D ________ Quad _____

E _________ Quad _____

F ________ Quad _____

A

NAMING POINTS

The Coordinate Plane

Steps to Make a Scatter Plot:1. Determine what will be x and y.

I. x – is in charge, it changes automaticallyII. y – depends on x, is not automatic

2. Determine units of each axis and label.I. Find range of variableII. Divide range by number of squaresIII. Always round up to “nice” unit

3. Plot points.

Make a Scatter PlotExampleThe age (in years) of seven used cars and the price (in thousands of dollars) paid for the cars are recorded in the table. Make a scatter plot and explain what it indicates.

0 1 2 3 4 5 6 78 9 1011 121314150

1234

56

78910

1112

131415

age of carp

rice

in

$1

,00

0

Age 4 5 3 5 6 4 7

Price6.9

6.1

7.5

5.2

4.2

7.1

3.0

How much would a 2-year old car cost?

Make a Scatter PlotExampleThe amount (in millions of dollars) spend in the United States on snowmobiles is shown in the table. Make a scatter plot and explain what it indicates.Year 90 91 92 93 94 95 96

Spent 322

362

391

515

715

924

970

LESSON 4.1 DHQ

• NONE TODAY

Tonight’s Homework Assignment:Coordinate WORKSHEET

WARM-UP Solve each equation for y.1) 2x + y = 10 2) 6x – 3y = -3

Find the value of y when x = -3.3) y = x – 7 4) y = -5x + 1

4.2 – GRAPHING LINEAR EQUATIONS

OBJECTIVES• Graph a using a table.• Graph and lines.

linear equation

horizontal vertical

Note

(1) All the Equations in Chap 4 refer 2 variable Linear Equations.

(2) The graph of each linear equation is a LINE.

SOLUTION OF AN EQUATIONIt is an ordered pair (x, y) that makes an equation true.

Example: x + 3y = 6 Is (-3, 3) a solution?

IS (-3, 3) THE ONLY SOLUTION?

x + 3y = 6

In pairs, try to come up with other solutions to the equation!

Try to come up with at least 2 more solutions.

x + 3y = 6

SOLUTIONS OF THE EQUATION

Solutions:

What would it look like if we plotted the solutions of the equation on a coordinate plane?

SOLUTIONS TO A LINEAR EQUATION

All the that lie on the line are the solutions to the !

points

equation

How many solutions did we have in our previous example?

HOW TO CHECK IF A POINT IS A SOLUTION

Method 1: Using a Graph• Check to see if the point is on the line.

Is (3, 1) a solution of the equation 2x – y = 5?

Is (0, -4) a solution of the equation 2x – y = 5?


HOW TO CHECK IF A POINT IS A SOLUTION

Method 2: Checking algebraically• Plug the point into the equation and see if it

is true.Is (3, 1) a solution of the equation 2x – y = 5?

Is (0, -4) a solution of the equation 2x – y = 5?


BREAK TIME

Use this time to relax, stretch out, talk to a neighbor, or try the following rebus puzzles.

USING A TABLE TO GRAPH AN EQUATIONSteps:1. Rewrite the equation so that it

says “y = …” – (called function form)

2. Make a table and choose values for x.

3. Plot the points on a coordinate plane and graph with a straight line.

USE A TABLE TO GRAPH THE FOLLOWING EQUATION

y + 1 = 2xStep 1: Rewrite the equation so that it says “y

= …”

Step 2: Make a table and choose values for x. x y

USE A TABLE TO GRAPH THE FOLLOWING EQUATION

y + 1 = 2xStep 3: Plot the points on a coordinate plane and graph with a straight line.

x y

PAIRS PRACTICEGraph y + 2 = 3x.

x y

PAIRS PRACTICEGraph y + 3 = x.

x y

GRAPHING HORIZONTAL AND VERTICAL LINES

Horizontal = left to right Vertical = up and down

x = any numbery = any number

MEMORIZE THIS!

vertical line

horizontal line

HORIZONTAL LINES

GRAPH THE FOLLOWING EQUATION

y = 5

Horizontal line where every y-coordinate on the line is 5.


y = -1

VERTICAL LINES


x = -3

Vertical line where every x-coordinate on the line is -3.


x = 5

LESSON 4.2 DHQ

Decide whether the given ordered pair is a solution of 2x – 3y = 8.

a. (-2, -4) b. (7, -2)

Rewrite 4x – 2y = 18 in function form.

Tonight’s Homework Assignment:Page/s: 214-215#’s 15-20, 30-32, 36-37, 60

4.3 - QUICK GRAPHS USING INTERCEPTS

Objectives:Find the intercepts of a graph of a linear equation.

1

2

3

5

6

4

7

-6

-2

-4

1 2 3 4 5 6 7-6 -4 -2

HOW MANY POINTS DOES IT TAKE TO DETERMINE A LINE?

We need at least two points to determine a line.

INTERCEPTS

x-intercept – the point where a line or curve crosses the x-axis. This is always written as (x, 0).

y-intercept – the point where a line or curve crosses the y-axis. This is always written as (0, y).

INTERCEPTS

1

2

3

5

6

4

7

-6

-2

-4

1 2 3 4 5 6 7-6 -4 -2

x-intercept?

y-intercept?

(0, 3)y = 3

(2, 0)x = 2

WHAT IF WE ARE NOT GIVEN A GRAPH? HOW DO WE DETERMINE THE X AND Y-INTERCEPTS?

WE CAN FIND THE…

x-intercept by setting y = 0

y-intercept by setting x = 0

FIND THE X-INTERCEPT AND Y-INTERCEPT OF THE GRAPH OF THE FOLLOWING EQUATION.

To find the x-intercept, set y = 0 and solve for x.

To find the y-intercept, set x = 0 and solve for y.

2x + 3y = 6

FIND THE X-INTERCEPT AND Y-INTERCEPT. THEN GRAPH THE LINE.

1

2

3

5

6

4

7

-6

-2

-4

1 2 3 4 5 6 7-6 -4 -2

2x – y = 4


1

2

3

5

6

4

7

-6

-2

-4

1 2 3 4 5 6 7-6 -4 -2

y – 2x = 3

BREAK TIME Use this time to relax, stretch out, talk to a

neighbor, or try the following rebus puzzles.


1

2

3

5

6

4

7

-6

-2

-4

1 2 3 4 5 6 7-6 -4 -2

y = 2x + 4

GRAPH AND WRITE THE EQUATION OF THE HORIZONTAL LINE PASSING THROUGH (3, -4) AND (-6, -4). 1

2

3

5

6

4

7

-6

-2

-4

1 2 3 4 5 6 7-6 -4 -2

GRAPH AND WRITE THE EQUATION OF THE VERTICAL LINE PASSING THROUGH (3, 2) AND (3, -5).

1

2

3

5

6

4

7

-6

-2

-4

1 2 3 4 5 6 7-6 -4 -2

RECAP:

1. The x-intercept and y-intercept are the points at which a line or curve cross the x and y-axis, respectively.

2. To find the x-intercept, set y = 0.3. To find the y-intercept, set x = 0.4. We can graph a line by connecting

the two intercepts.

LESSON 4.3 DHQ

1. Give the x- and y- intercepts of the graph of 2x – y = -4.

2. Graph 2x – 3y = 6. Label the points where the line crosses the axes.

Tonight’s Homework Assignment:

Page/s: 221-222#’s 35-37, 44-49, 56-57• QUIZ Friday

4.4 – SLOPE OF A LINE

Objective:Find the slope of linear equations.

WHAT IS SLOPE?

Have you ever…

walked up a ramp?skied down a hill?jumped to dunk a ball?

What are some other examples?

Mathematical Definition Slope (m): Number of units the line rises or falls for each unit of horizontal change from left to right.

m = riserun

vertical change

horizontal change

What is the slope?

IF YOU KNOW TWO POINTS, SLOPE IS GIVEN BY THE EQUATION…

2 1

2 1

y ym

x x

WARNING!!! Important!

FIND THE SLOPE OF THE POINTS (-2, 2) & (3, 4).

Check:

2 1

2 1

y ym

x x

FIND THE SLOPE OF THE POINTS (-1, 2) & (3, 2).

2 1

2 1

y ym

x x

HORIZONTAL SLOPES

THE SLOPE OF A HORIZONTAL LINE IS

ALWAYS ZERO!

m = riserun

What is the rise of any horizontal

line?

FIND THE SLOPE OF THE POINTS (2, 4) & (2, 1).

2 1

2 1

y ym

x x

VERTICAL SLOPES

THE SLOPE OF A VERTICAL LINE IS ALWAYS UNDEFINED!

m = riserun

What is the run of any horizontal

line?

Check: Try to divide any number by zero!

SKIING ANALOGYPositive Slope = Going up the chair

lift. The height increases.Negative Slope = Going down the hill.

The height decreases.Horizontal Slope = The height on the

ground is zero and stays the same as you move.

Undefined Slope = Most skiers that look at a hill that drops would say this is impossible!

FIND THE SLOPE PASSING THROUGH THE

POINTS!

(4, 5) and (2, 3)Check:


POINTS!(2, 3) and (-3,

0)Check:


POINTS!(-6, -1) and (-6,

4)Check:

FIND THE VALUE OF Y SO THAT THE LINE PASSING THROUGH THE POINTS (3, Y) AND (5, 8) HAS A SLOPE OF 2.

FIND THE VALUE OF Y.

(0, y), (2, 5)m = 2

LESSON 4.4 DHQ

Tonight’s Homework Assignment:

Page/s: 230-231

#’s 23-28, 35-36, *38

1. Find the slope of the line. Use formula to find the slope.

a. (-2, 2), (0, 4)

b. (1, 1), (4, 2)

2. Find the slope of the line.

LESSON 4.6 WARM-UP

Rewrite 4x – 2y = 18 in function form.

4.6 – QUICK GRAPHS USING SLOPE-INTERCEPT FORM

OBJECTIVE:Write and graph linear equations in slope-intercept form!

SLOPE-INTERCEPT FORMA linear equation written in the form of:

The slope of the line is m.

The y-intercept is b.

𝑚=𝑟𝑖𝑠𝑒𝑟𝑢𝑛

Where a line crosses the y-axis.

Organization Check:We can now graph an equation in three different ways:1. Using an x, y table2. Using x-intercepts, y-

intercepts3.Slope-intercept

form

WRITE THE EQUATION IN SLOPE-INTERCEPT FORM, THEN GRAPH.

𝑦=−𝑥+2Steps:1. Set equation to y=2. Identify the y-intercept (b)

and plot it on y axis3. Identify the slope (m) and

plot 1-2 additional points beginning at the y-int (b)

4. Draw your line, with arrows!

−3 𝑥+𝑦=7Write the equation in slope-intercept form, then graph.

3 𝑥−5 𝑦=15Write the equation in

slope-intercept form, then graph.


𝑦=−4


x

BREAK TIME! Use this time to get up, stretch out, relax, talk to

a neighbor, or try the following rebus puzzles…

PARALLEL LINES

Two different lines in the same plane are _______ if they do not intersect.

PARALLEL LINES

Lines are parallel if they have the same ______.

ARE THE LINES PARALLEL?

y = 2x + 3

y = 3 + 2x


y = -2x + 3

y = 3 + 2x


4x + 2y = 8

y = -2x + 2

LESSON 4.6 DHQ

1. Write x + y + 3 = 0 in slope-intercept form. Then graph the equation.

Tonight’s Homework Assignment:• DUE TOMORROW!

Page/s: 244-2456#’s 13-18, 29-31, 34-36, 46-48

• QUIZ TOMORROW – L 4.4/4.6 – STUDY!

WARM-UP 1. What is the slope of the line 5y = -3x

+10 ?a. -3 b. c. d. 2 e. 10

2. What is the lowest value of x that satisfies the equation x2 – 7x + 6 = 6 ?

a. -3 b. -6 c. 0 d. 6 e. 1

3

5

5

3

FUNCTIONS AND RELATIONS

Section 4.8

OBJECTIVE:

Identify when a relation is a function.

He knows how to do that, and soon you will

too!

REVIEW: What is a function?

A function is a rule that establishes a relationship with an input and an output.

inputoutpu

t

REMEMBER? For every input, there is exactly _____

output!one

IS IT A FUNCTION?

Input 0 1 1 3

Output 3 1 3 2

Domain ?Range ?

Input 0 1 1 3

Output 3 1 1 2

IS IT A FUNCTION?

Domain ?Range ?

Input 0 1 2 3

Output 3 4 5 6

IS IT A FUNCTION?

Domain ?Range ?

HOWEVER…There are rules that associate more than one output for each input. But, we can’t call it a function anymore. We call it a…

Relation

WHAT IS A RELATION?Any set of ordered pairs!

It doesn’t matter how many different outputs an input has!

WHAT WE NEED TO DO IS…

Decide if a relation (set of ordered points) is a function.

IS THE RELATION A FUNCTION?

Yes Yes No

TAKE A MINUTE TO…

• Complete #1-3 on your in-class practice

• Read the instructions carefully• If you have a question, raise your hand

BESIDES LOOKING AT A TABLE AND DECIDING, WE ARE GOING TO LOOK AT GRAPHS AS WELL!

Vertical Line Test – a relation is a function, if NO vertical line passes through two or more points!

Function

Relation – Not a

function

Is the relation a function?

Is the relation a function?

Relation – Not a

function

TAKE A MINUTE TO…



FUNCTION NOTATION F(X)

• f(x) means the value of f at x.

• You can replace it with y.

• Does not mean f times x.It means f at x.

EVALUATE THE FUNCTION FOR THE GIVEN VARIABLE.

f(x) = 2x – 3 when x = -2, 0, and 3

Steps:1. Write the original function.2. Substitute -2 for x.3. Simplify.

g(x) = -5x + 3 when x = -3, 0, and 1

EVALUATE THE FUNCTION FOR THE GIVEN VARIABLE.

TAKE A MINUTE TO…



GRAPH THE FUNCTION!

Graph f(x) = -2x + 3Steps for using slope-intercept form:1. Rewrite the function as “y = …” 2. Find y-intercept and slope.3. Graph and connect.

GRAPH THE FUNCTION!

Graph f(x) = x + 4Steps for using intercepts:1. Find the x-intercept by

setting y = 0.2. Find the y-intercept by

setting x = 0.3. Plot the intercepts on the

coordinate plane and graph the line.

WITH YOUR PARTNER, YOU NEED TO GRAPH THREE FUNCTIONS!

Next to graphs done by slope-intercept (2 done by this method)

Next to graphs done by using x and y-intercepts (1 done by this method)

HOMEWORK pgs. 259-260 #16-17, 20-21, 37-40 Chapter 4 Test Next Week!

w elcome to c hapter 4!. psae p ractice 1. you are the new maintenance person for the local school...

Documents