Unit 4 Analyzing Linear Equations - Aliquippa School … Manual42.pdf131 Unit 4 – Analyzing Linear Equations 4–1 Slope 4–2 Slope Intercept Equation 4–3 Point Slope Equation

131

Unit 4 – Analyzing Linear Equations

4–1 Slope

4–2 Slope Intercept Equation

4–3 Point Slope Equation

4–4 Parallel/Perpendicular Lines

4–5 Scatter Plots

132

Review Question How did we find the slope of the equations in the previous unit?

The change in y, over the change in x.

Today, we will try to figure out why that works.

Discussion How would you compare these two sled riding hills? The first hill is steeper

In math, we don’t use “steep”. Instead we use the word slope.

What is causing the first hill to be steeper than the second?

The amount that the y value changes is bigger than the amount that the x value changes. Show this

in the drawings above. So to calculate how steep a line is (slope), we must compare the changes in y

and x.

How do you find the change in height (y)? Subtract

How do you find the change in horizontal distance (x)? Subtract

How would you compare the slopes (steepness) of line 1 and 2?

The steepness is the same but the direction is different. Therefore, lines 1 and 2 have the same numeric

slope but line 2 is negative. Notice the negative doesn’t have anything to do with the steepness of the

line, but purely the direction of the line.

A positive slope creates a line that goes up/right.

A negative slope creates a line that goes down/right.

Slope – steepness and direction of a line

Section 4-1: Slope (Day 1)

y

x

y

x

2 1

133

SWBAT calculate the slope of a line base on a graph

Example 1: Find the slope of the line.

m = 2/6

Example 2: Find the slope of the line.

m = -5/4

Example 3: Draw a line that has a slope of 2

7.

What did we learn today?

2

6

5

2

7

4

134

Find the slope of the line.

1. 2.

3. 4.

5. Change in y: -8 in 6. Change in x: 3 feet

Change in x: 5 in Change in y: 4 feet

7. Change in y: 8 in 8. Change in y: 300 cm

Change in x: 2 feet Change in x: 4 m

9. Draw a line that has a slope of 2

5. 10. Draw a line that has a slope of

3

2 .

Section 4-1 In-Class Assignment (Day 1)

5 1

2 6

3 ft

24 in

135

Review Question What does slope mean? Steepness and direction of a line

How do you find slope? Compare the changes in y’s and x’s

How do you define direction? Positive – up/right, Negative – down/right

Discussion So to calculate how steep a line is (slope), we must compare the changes in y and x.

How do you find the change in height (y)? Subtract

How do you find the change in horizontal distance (x)? Subtract

Therefore, to calculate the between two points:

SWBAT calculate the slope given two points

Example 1: Find the slope between (4, 12) (2, 1). 11/3

What direction does the line go? Up/Right

Graph the two points to confirm answer.

Example 2: Find the slope between (-4, 1) (2, 2). -1/-6 = 1/6

What direction does the line go? Up/Right


Example 3: Find the slope between (2, 1) (-1, 8). -7/3

What direction does the line go? Down/Right


You Try! Calculate the slope and direction of the line. Then graph the two points.

1. (4, 8) (3, 2) 6/1, Up/Right

2. (1, 5) (7, 4) 1/-6, Down/Right

3. (-1, 2) (1, -4) 6/-2, Down/Right

4. (-2, 1) (3, 8) 7/1, Up/Right



xx

yym

136

For each problem:

a. Find the slope.

b. Describe the line as up/right or down/right.

c. Graph the two points.

1. (6, 8) (2, 7)

2. (8, 8) (6, 1)

3. (2, 6) (3, 1)

4. (-4, -8) (1, 4)

5. (10, 5) (-4, 1)

6. (-2, 1) (3, -5)

Find the slope.

7. 8.

Section 4-1 Homework (Day 2)

4 ft

36 in

137


How do you find slope? Compare the changes in y’s and x’s

How do you define direction? Positive – up/right, Negative – down/right

Discussion We know that if the y’s are bigger than the x’s the line is steep. We also know that if the x’s are bigger

than the y’s the line is flat.

What if the y’s and x’s are the same? It would give us a slope of 1

What kind of line would that give us? “Average” Estimate the slope of each of the lines starting with the middle line.

m = 4/0 m = 4/2 = 2 m = 2/2 = 1 m = ¼ = .25 m = 0/4 = 0

How steep is a vertical line? It is so steep that we can’t put a number on it. It is undefined.

How steep is a horizontal line? It is so flat that it is zero.

SWBAT calculate the slope of horizontal and vertical lines

Definitions Horizontal Line – slope of zero

Vertical Line – undefined slope

Example 1: Find the slope between (4, 5) (4, 8). Then graph. m = 3/0 = Undefined (vertical)

Example 2: Find the slope between (5, 3) (2, 3). Then graph. m = 0/3 = 0 (horizontal)

You Try! Calculate the slope and direction. Then graph.

1. (6, 8) (4, 8) m = 0

2. (7, 5) (7, 4) m = undefined

3. (1, 2) (3, 4) m = 1

4. (8, 4) (3, 3) m = 1/5


2

2

4

2 4

1

138


For each problem:

a. Graph the two points.

b. Find the slope.

c. Describe the line as up/right, down/right, horizontal, or vertical.

1. (6, 2) (2, 1)

2. (4, 2) (3, 1)

3. (1, 6) (3, 8)

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

5. (4, 6) (-3, 6)

6. (2, 0) (0, 8)

7. (8, 3) (2, 4)

8. (-4, 1) (0, 2)

9. (-1, 6) (4, 6)

10. (3, 6) (3, 8)

Find the slope of the line.

11. 12.


10

12

139


Discussion We are going to learn the slope intercept equation today.

What two things do you think we need to know to use this equation? Slope, intercept

What does intercept mean? The place where a line touches an axis.

What does the y-intercept mean? Place where line touches the y axis

What is the y intercept?

1. 2. 3. (5, 2) (0, -1)

y-int = -1

y-int = -3 y-int = 1

SWBAT write an equation of a line using the slope intercept equation

Definitions x-intercept – place where line touches the x-axis

y-intercept – place where line touches the y-axis

y = mx + b (Slope intercept equation)

m = slope

b = y-intercept

Example 1: Write an equation of a line with a slope of -2 and a y-intercept of 8. y = -2x + 8

Example 2: Write an equation of a line that goes through the point (0,-4) and has a slope of 3

1.

y = 1/3x – 4

Example 3: Write an equation of a line that goes through the points (0,3) and (5,1).

What two things do we need to know in order to write an equation of a line? Slope, Intercept

m = 2/-5; y = -2/5x + 3

Section 4-2: Slope Intercept Equation (Day 1)

140

(-5, 2)

(0, -2)

You Try! Write an equation with the following conditions.

1. m = 4, y-int = -2 y = 4x – 2

2. Horizontal line that touches the y-axis at -3. y = 0x – 3; y = -3

3. (0, -1) (3, -2) y = -1/3x – 1

4. Vertical line that touches the x-axis at 4. x = 4

5. Write an equation of a line that goes through the point (0, 2) and has a slope of 5

1. y = 1/5x + 2


Write an equation of the line with the given conditions.

1. Slope: 2, y-intercept: -6

2. Slope: -3, y-intercept: 5

3. Horizontal line that touches the y axis at 2.

4. Slope: 2

1, y-intercept: 3

5. (0, 4) (3, -1)

6. Slope: 5

3 , y-intercept: 0

7. Slope: -1, y-intercept: -6

8. A line that goes through the point (0, -3) and has a slope of 5

1.

9. Slope: 0.5, y-intercept: 7.5

10. Vertical line that touches the x axis at -1.

Write an equation of the line shown in each graph.

11. 12.


141

Review Question What is the slope intercept equation? y = mx + b

What does each letter represent?

y is y

m is slope

x is x

b is the y-intercept

Discussion Why is this equation so easy? You just plug in the slope and y-intercept.

How could this equation help us? Graph

SWBAT graph an equation of a line using the slope intercept equation

Example 1: Graph: y = 3x + 1

How would you graph this line in previous units? T-Charts

How could you graph this line using the slope intercept equation?

Start at (0, 1) because that is the y-intercept then go up three and over one using the slope.

Which way is easier? Slope Intercept

Example 2: Graph: y + 2x = -4

What is different about this equation? It’s not in the slope intercept form.

After some manipulation: y = -2x – 4

To graph start at (0, -4) because that is the y-intercept then go down two and over one using the

slope.

Example 3: Graph: 24

3 xy

To graph start at (0, -2) because that is the y-intercept then go up three and over four using the

slope.

Example 4: Graph: 2x – 5y = -8

What is different about this equation? It’s not in the slope intercept form.

After some manipulation: 5

8

5

2 xy

To graph start at (0, 8/5) because that is the y-intercept then go up two and over five using the

slope.


142

You Try! Graph each line.

1. y = 2x – 5 Start at (0, -5) then go up 2 over 1

2. 55

2 xy Start at (0, 5) then go down 2 over 5

3. y – 4x = 1 Start at (0, 1) then go up 4 over 1

4. x = -2 Vertical line at x = 2

5. 4x – 3y = 5 Start at (0, -5/3) then go up 4 over 3

6. y = 5 Horizontal line at y = 5


Graph each equation.

1. y = 3x + 1 2. y = x – 2

3. y = -4x + 1 4. y = -x + 2

5. 42

1 xy 6. 3

3

1 xy

7. y + 3x = -2 8. y – 2x = -3

9. y = 2x + 3 10. y = -5x + 1

11. y + x = 3 12. 55

2 xy

Write the equation of the line. Then graph.

13. Horizontal line that touches the y axis at 2.

14. Slope: 3

2 , y-intercept: 2

15. (0, -1) (3, 4)

16. A line that goes through the point (0, 5) and has a slope of 7

2.

17. Write an equation of a line that passes through the origin with slope 3.

18. (3, 9) (0, 6)


143


What does each letter represent?

y is y, m is slope, x is x, b is the y-intercept

Discussion Why do we learn about linear equations?

The writing of linear equations is used often in computer programming.

Think about the automated system in a parking garage. A computer reads your ticket. Then it calculates

how long you were parked. Finally, it calculates how much you need to pay.

What type of calculations/equations does it use? First, it must subtract the time you entered from the time you left the garage. Then it calculates

how much you will pay.

Does the amount you pay increase each minute you are parked?

No, it increases based on increments of time

Is this a linear relationship? No

If it is not linear, then what would it look like?

Notice this is also applicable to fantasy football. A player gets 1 point for every ten yards he rushes for.

Therefore, he gets 1 point for 10 yards up to 19 yards. Once he gets to 20 yards, it jumps to 2 points.

This is what we will be doing today. We will take a real life situation and try to write and Algebraic

equation for it.

SWBAT write a linear equation given a real life situation

Example 1: At a bowling alley, it costs $3 to rent shoes plus $2 per game.

Define variables. Then write an equation to model this situation.

Let T = total cost

Let g = number of games

T = 2g + 3


144

How much does it cost for 4 games?

T = 2(4) + 3

T = 8 + 3

T = $11

How many games did I play if it cost $15?

15 = 2g + 3

-3 - 3

12 = 2g

2 2

6 = g


1. Define variables. Then write an appropriate equation.

a. Jimmy rented a bike for $10 plus $2 per hour. C = 10 + 2h

b. An auto repair shop charges $50 plus $25 per hour. C = 50 + 25h

c. A candle is 6 inches tall and burns at a rate of 2

1inch per hour. H = 6 – 1/2h

d. The temperature is 20° and is expected to fall 2° each hour during the night. T = 20 – 2h

2. Your grade is a 78% and increases 3% for each assignment you turn in.

a. Define variables. Then write an appropriate equation. G = 78 + 3a

b. How many assignments do you have to do in order to get an 87%? 3

c. How many assignments do you have to do in order to get an 85%? 3

d. What will your grade be if you do 7 assignments? 99%

Section 4-2 In-Class Assignment (Day 3)

145

1. Define variables. Then write an appropriate equation.

a. You have $100 and plan to save $5 per week.

b. You lost 10 pounds and plan to lose 3 pounds per month.

c. You make $25000 per year and expect to get a raise of $3000 per year.

2. A barber charges $8 for a haircut. He must pay the owner of the barber shop $50 per day for the use

of one of the chairs.

a. Define variables. Then write an appropriate equation.

b. How much does he make for 24 haircuts?

c. How much does he make for 5 haircuts?

d. How many haircuts did he do if he made $30?

e. How many haircuts does he have to give to break even?

3. You are $9000 in debt. You plan to pay back $1200 per year.

a. Define variables. Then write an appropriate equation to model this situation.

b. How much will you owe after 3 years?

c. How many years have you been paying if you still owe $3000?

d. How many years will it take to be out of debt?


146


When do we use the slope intercept equation?

When we are given a slope and y intercept. That is why it is called the slope intercept equation.

Discussion How would write an equation of a line that passes through (-3, 1) and has a slope of -2?

You can’t right now

Why can’t we use the slope intercept equation? Don’t know the intercept

Can you guess what the name of the equation we need?

Point Slope; because they give us a point and a slope

SWBAT write a linear equation using the point slope equation

Definition Point Slope equation: y – y1 = m(x – x1)

y = y

x = x

x1, y1 = coordinates of the point

m = slope

Example 1: Write an equation of a line that passes through (-3, 1) and has a slope of -2.

y – 1 = -2(x + 1)

y = -2x – 1

Example 2: Write an equation of a line that passes through (1, 2) and (3, 6).

What do we need to use the point slope equation? Point, slope

m = 4/2 = 2

y – 2 = 2(x – 1)

y = 2x

Notice we can use either point.

Notice we can use point slope when we have two points.

You Try! Write an equation of a line in slope intercept form with the following conditions.

1. Passes through (-3, 2), m = 4 y = 4x + 1

2. Passes through (2, 4) and (3, -1) y = -5x + 14

3. m = 3

1 , y-int = -2 y = -1/3x – 2

4. Passes through (2, -1), m = -3 y = 3x – 7

Section 4-3: Point Slope Equation (Day 1)

147


Write an equation of a line in slope intercept form with the following conditions. Then graph.

1. m = 2, (2, 3) 2. m = -3, (-2, 1) 3. (5, 2) (4, 5)

y = 2x – 1 y = -3x – 5 y = -3x + 17

4. m = 3

2 , y-int = 2 5. (4, 2) (5, 4) 6. m = -2, (-1, -5)

y = -2/3x + 2 y = 2x – 6 y = -2x – 7

7. m = 5

1, y-int = 4 8. m = -4, (-1, -2) 9. (-2, 5) (-4, 9)

y = 1/5x + 4 y = -4x – 6 y = -2x + 1

10. m = 1, y-int = -2 11. (5, -2) (3, -5) 12. m = 4

1, (2, -1)

y = x – 2 y = 3/2x – 19/2 y = 1/4x – 3/2


148


What is the point slope equation? y – y1 = m(x – x1)

How do you know when to use each one? Look at the name of the equation

Discussion When do we use the slope intercept equation? When we have a slope and y-intercept

When do we use the point slope equation? When we have a point and slope or when we have 2 points

SWBAT write a linear equation using the point slope equation and slope intercept equation

Example 1: Given y = -2x + 5. Find the slope and y-int. Then graph.

m = -2

y-int = 5

Graph: Start at (0,5) then go down 2 and over 1.

Example 2: Given a line passes through (2, -4) and has a slope of 3. Find the slope and y-int. Then

graph.

y + 4 = 3(x – 2)

y = 3x – 10

m = 3

y-int = -10

Graph: Start at (0, -10) then go up three and over one.

You Try! For each problem do the following:

a. Write the equation in slope intercept form

b. Find the slope

c. Find the y intercept

d. Graph

1. y = -2x + 3 y = -2x + 3, m = -2, y-int = 3

2. Passes through (-1, 3), m = 2 y = 2x + 5, m = 2, y-int = 5

3. x = 4 m = undefined, y-int = DNE

4. Passes through (3, 8) and (4, 5) y = -3x + 17, m = -3, y-int = 17

5. y = -2 y = -2, m = 0, y-int = -2



149

For each problem do the following:

a. Write the equation in the slope intercept form

b. Find the slope


d. Graph

1. y = 4x + 1 2. (6, 2) (4, 8)

y = 4x + 1, m = 4, y-int = 1 y = -3x + 20, m = -3, y-int = 20

3. m = -3, y-int = 2 4. y = 3

y = -3x + 2, m = -3, y-int = 2 y = 3, m = 0, y-int = 3

5. x = - 4 6. m = -3, (2, -1)

m = undefined, y-int = none y = -3x + 5, m = -3, y-int = 5

7. y = -2x + 5 8. (2, -5) (-2, 7)

y = -2x + 5, m = -2, y-int = 5 m = undefined, y-int = none

9. m = 4, (-3, 2) 10. (2, 5) (4, 5)

y = 4x + 14, m = 4, y-int = 14 y = 5, m = 0, y-int = 5


150


What is the point slope equation? y – y1 = m(x – x1)

How do you know when to use each one? Look at the name of the equation

Discussion What do we know about the y-intercept? It’s where a line touches the y-axis.

What do we know about the value of ‘x’ at a y-intercept? It is 0.

What do we know about the x-intercept? It’s where a line touches the x-axis.

What do we know about the value of ‘y’ at an x-intercept? It is 0.

How could you find the x-intercept? Put 0 in for ‘y’

SWBAT write a linear equation and identify the key components of the line

Example 1: Given y = 3x + 9.

Find the slope, y-int and x-int. Then graph.

m = 3

y-int = -3

Graph: Start at (0, 9) then go up 3 and over 1.

Example 2: Write an equation of a line that passes through (1, -3) and has a m = -2. Find the slope, y-int

and x-int. Then graph.

y + 3 = -2(x – 1)

y = -2x – 1

m = -2

y-int = -1

x-int = -1/2

Graph: Start at (0, -1) then go down 2 and over 1.

You Try! For each problem do the following:

a. Write the equation in slope intercept form

b. Find the slope


d. Find the x intercept

e. Graph

1. Passes through (-3, 2), m = -2 y = -2x – 4, m = -2, y-int = -4, x-int = -2

2. y + 4x = 12 y = -4x + 12, m = -4, y-int = 12, x-int = 3

3. 3

1x y = ?, m= und, x-int = -1/3, y-int = DNE

4. Passes through (3, 4) and (5, 6) y = 2x – 2, m = 2, y-int = -2, x-int = 1

5. m = 3, y-int = -1 y = 3x – 1, m = 3, y-int = -1, x-int = 1/3


151


For each problem do the following:

a. Write the equation in the slope intercept form

b. Find the slope


d. Find the x intercept

e. Graph

1. y = -2x + 4 2 (1, 5) (2, 3)

y = -2x + 4, m = -2, y-int = 4, x-int = 2 y = 2x + 3, m= 2, y-int = 3, x-int = -3/2

3. m = 2, y-int = -2 4. y = -5

y = 2x – 2, m = 2, y-int = - 2, x-int = 1 y = - 5, m = 0, y-int = -5, x-int = none

5. x = 2 6. m = 3, (-2, 3)

x = 2, m = undefined, y-int = none, x-int = 2 y = 3x + 9, m = 3, y-int = 9, x-int = -3

7. 3

1

4

1 xy 8. (4, -6) (3, -2)

y = 1/4x + 1/3, m = 1/4, y-int = 1/3, x-int = -4/3 y = -4x + 10, m = -4, y-int = 10, x-int = 10/4

9. m = -2, (3, -5) 10. (-2, 5) (4, 5)

y = -2x + 1, m = -2, y-int = 1, x-int = 1/2 y = 5, m = 0, y-int = 5, x-int = none


152

Review Question What is the slope of a horizontal line? 0

What is the slope of a vertical line? Undefined

Discussion What does parallel mean? Don’t touch

What makes two lines parallel? Same slopes

SWBAT write an equation of a line that is parallel to another line

Example 1: Write an equation of a line in slope intercept form that is parallel to y = 4x + 2 and goes

through the point (4, 1). Let’s take a look at what they are asking first.

Now let’s try to write the equation. Remember we need a point and a slope in order to write an equation

of a line. What is the slope of our new line? 4

y – 1 = 4(x – 4)

y = 4x – 15

Example 2: Write an equation of a line in slope intercept form that is parallel to y + 2x = 3 and goes

through the point (-2, 5). Let’s take a look at what they are asking first.


of a line. What is the slope of our new line? -2

y – 5 = -2(x + 2)

y = -2x + 1

You always ask when can we use this? This can helpful when you are building cement steps. The forms

to the steps must be parallel to each other in order to ensure that the steps are flat.

Section 4-4: Parallel/Perpendicular Lines (Day 1)

153

You Try! Write an equation of a line in slope intercept form that is parallel to each line and goes through the given

point.

1. y = 2x – 3; (-1, 3) y = 2x + 5

2. y + 3x = 4; (2, -3) y = -3x + 3

3. x = 3; (2, 4) x = 2

4. 3x + 2y = 5; (1, 5) y = -3/2x + 6 1/2

5. y = 5; (-1, 2) y = 2


Write an equation of a line in slope intercept form that passes through the given point and is

parallel to the given line. Then state the slope, y-intercept, and x-intercept. Then graph the

original line and new line.

1. y = 3x + 5; (3, 7) 2. y = -x – 1 ; (1, -5)

y = 3x + 2, m = 3, y-int = 5, x-int = 2/3 y = -1x – 1, m = -1, y-int = -1, x-int = -4

3. y + 4x = 3; (-2, -5) 4. y = 4; (2, -3)

y = -4x – 13, m = -4, y-int = -13, x-int = 13/-4 y = -3, m = 0, y-int = -3, x-int = none

5. 52

1 xy ; (2, 3) 6. 2y + 5x = -3; (-2, 1)

y = 1/2x + 2, m = 1/2, y-int = 2, x-int = -4 y = -5/2x – 4, m = -5/2, y-int = -4, x-int = -8/5

7. x = -3; (1, -5) 8. y = -2x – 1; (1, 2)

x = 1, m = undefined, y-int = none, x-int = 1 y = -2x + 4, m = -2, y-int = 4, x-int = 2

9. y = 4x; (1, 2) 10. y = -2x – 5; (0, 3)

y = 4x – 2, m = 4, y-int = -2, x-int = 1/2 y = -2x + 3, m = -2, y-int = 3, x-int = 3/2


154

Review Question What is the slope of a horizontal line? 0

What is the slope of a vertical line? Undefined

Discussion What does perpendicular mean? Intersect at a right angle

What makes two lines perpendicular? Opposite reciprocal slopes

SWBAT write an equation of a line that is perpendicular to another line

Example 1: Let’s make sure we know how to find perpendicular slopes:

a. m = 3

2, Perpendicular Slope = ? -3/2

b. m = 4

1 , Perpendicular Slope = ? 4

c. m = 8, Perpendicular Slope = ? -1/8

d. Horizontal Line, Perpendicular Slope = ? Undefined

e. Vertical Line, Perpendicular Slope = ? Zero

Example 2: Write an equation of a line in slope intercept form that is perpendicular to y = 4x – 2 and

goes through the point (5, -2). Let’s take a look at what they are asking first.


of a line. What is the slope of our new line? -1/4

)5(4

12 xy

4

3

4

1 xy


155

Example 3: Write an equation of a line in slope intercept form that is perpendicular to 32

1 xy and

goes through the point (-3, 2).

)3(22 xy

82 xy


1. y = 3x + 1; (1, -2); Perpendicular y = -1/3x – 1 2/3

2. 63

1 xy ; (-2, 3); Perpendicular y = -3x – 3

3. y = 5; (1, -1); Perpendicular x = 1

4. 2x + 5y = 1; (-1, 4); Parallel y = 2/5x + 3 2/5

5. x = 5; (-1, 2); Perpendicular y = 2


Write an equation of a line in slope intercept form that passes through the given point and is

perpendicular to the given line. Then state the slope, y-intercept, and x-intercept. Then graph the

original line and new line.

1. 53

1 xy ; (2, -3) 2. y = 3x – 1; (1, -3)

y = -3x + 3, m = -3, y-int = 3, x-int = 1 y = -1/3x – 2 2/3, m = -1/3, y-int = - 2 2/3, x-int = - 8

3. y + 2x = -1; (-2, -4) 4. y = 4; (-1, 3)

y = 1/2x – 3, m = 1/2, y-int = -3, x-int = 6 x = 1, m = undefined, y-int = none, x-int = -1

Write an equation of a line in slope intercept from with the following conditions. Then state the

slope, y-intercept, and x-intercept.

5. y = 3x – 4; (3, 1); Parallel 6. 17

1 xy ; (-3, 1); Perpendicular

y = 3x – 8, m = 3, y-int = -8, x-int = 8/3 y = 7x + 22, m = 7, y-int = 22, x-int = -22/7

7. x = -1; (3, -2); Perpendicular 8. 2x – y = 5; (1, -3); Parallel

y = - 2, m = 0, y-int = -2, x-int = none y = 2x – 5, m = 2, y-int = -5, x-int = 5/2


156

Review Question What makes two lines parallel? Same slopes

What makes two lines perpendicular? Opposite reciprocal slopes

Discussion How do you get better at something? Practice

Today, is going to be a day of practice.

SWBAT write an equation of a line that is perpendicular/parallel to another line

Example 1: Write an equation of a line in slope intercept form that is perpendicular to y = 3x – 2 and

goes through the point (-3, 2).

)3(3

12 xy

13

1 xy


1. y = 3x + 3; (1, -2); Parallel y = 3x – 5

2. 13

1 xy ; (4, -1); Perpendicular y = -3x + 11

3. y = 2; (4, -1); Parallel y = -1

4. y – 4x = 1; (-1, -2); Perpendicular y = -1/4x – 2 1/4

5. y = -4; (-1, 6); Perpendicular x = -1

6. 3y + 2x = 5; (-3, 2); Parallel y = -2/3x



157

Write the equation in slope intercept form then state the slope, y-intercept, x-intercept, and graph.

1. Write an equation of a line in slope intercept form that passes thru (2, 5) and (3, 2).

y = -3x + 1, m = -3, y-int = 1, x-int = 1/3

2. Write an equation of a line in the slope intercept form that has a slope of 3

1and passes through (2, -4).

y = 1/3x – 4 2/3, m = 1/3, y-int = 4 2/3, x-int = 14

3. Write an equation of a line in slope intercept form that is parallel to y = -2x + 5 and passes through

(5, -7).

y = -2x + 3, m = -2, y-int = 3, x-int = 3/2

4. Write an equation of a line in slope intercept form that is perpendicular to -2y – 8x = 2 and passes

through (8, -4).

y = 1/4x – 6, m = 1/4, y-int = -6, x-int = 24

5. Write an equation of a horizontal line that passes through the point (-2, 5).

y = 5, m = 0, y-int = 5, x-int = none

6. Write an equation of a line in slope intercept form that is perpendicular to the x-axis and passes

through (-7, -11).

x = 7, m = undefined, y-int = none, x-int = -7

7. Write an equation of a line in slope intercept form that is parallel to y = 2 and passes through (-4, 5).

y = 5, m = 0, y-int = 5, x-int = none

8. Write an equation of a line in slope intercept form that has a slope of 3

2 and passes through (0, -6).

y = -2/3x – 6, m = -2/3, y-int = -6, x-int = -9


158

Review Question How are perpendicular lines related? Their slopes are opposite reciprocals.

Discussion Let’s say that we collected some data on the number of hours we studied and the grade we received.

When plotted on a graph, would the data points be in a straight line? No

What are some things that would cause the data to be erratic? Difficulty of material, student

SWBAT read and create scatter plots

Definition Scatter plot – graph that shows the relationship of two sets of data

To find how many pieces of data are in a scatter plot, just count the dots!

Example 1: What was the highest score in the class? 95%

How many students studied for one hour? 4

How many students scored above 75%? 12

Example 2: Draw a scatter plot of the following data set. Time spent at the mall and amount of money

left.

Time Money

1 $200

3 $155

4 $145

7 $85

8 $80

9 $5

Section 4-5: Scatter Plots (Day 1)

Time Studying vs. Test Score

60

65

70

75

80

85

90

95

100

0 1 2 3 4

Time studying (Hours)

Tes

t S

core

(P

erce

nt)

Test score

Time vs. Money

0

50

100

150

200

250

0 2 4 6 8 10

Time (hours)

Am

ou

nt

($)

159

Things to remember when creating a scatter plot:

1. Choose good starting and ending points for each axis

2. Choose sensible scales

3. Time always goes on the x - axis

You Try! 1. Create a scatter plot based on the following sets of data.

Time Money

Earned

1 $8

2 $15

4 $35

5 $45

6 $50

9 $85

2. Create a scatter plot based on the following sets of data.

Age of

Car

Value of

Car

1 $15,000

2 $12,500

3 $10,000

6 $7,500

10 $5,000

15 $375

Discussion How does this relate to our study of lines? The data is almost in a line pattern.

Notice the data sets are sort of in a line pattern. Tomorrow we will try to summarize these data sets by

drawing a line that best fits data sets.


160

1. The following data sets represent the amount of time students spend playing Playstation and their

average grades:

Time

(hours) 1 2 3 4 5 6 7 8

Average

Grade 96 98 85 83 78 81 68 65

a. Draw a scatter plot based on the data sets.

b. Predict the average grade for a student that plays for 15 hours.

c. Predict what the time would be when a student started to fail.

2. The following data sets represent the amount of time driving in a car and how far you traveled.

Time

(hours) 1 2 3 4 5 6 7 8

Distance

Traveled 45 105 140 210 270 325 385 420

a. Draw a scatter plot based on the data sets.

b. Predict the total distance traveled after 12 hours.

c. Predict the amount of time necessary to drive 800 miles.


161

Review Question Why couldn’t you use a pie graph or bar graph for the two homework problems? Two sets of data

When should we use a scatter plot? When we are graphing two sets of data

Discussion What is the difference between the two scatter plots in each of your two homework problems? Direction

What is causing this to happen? Data; slope

SWBAT identify positive/negative relationships

SWBAT draw a best fit line

Definitions Positive Relationship/Slope - up/right, x increases/y increases

Example: Driving Time and Distance

Can someone give me another example of a positive relationship?

What would happen if x and y were decreasing? It would still be positive. Think of a slope of a

negative over a negative. It is still a positive slope.

Negative Relationship/Slope - down/right, x increases/y decreases or vice versa

Example: Time and amount of gas left

Can someone give me another example of a negative relationship?

What type of relationship exists in homework examples 1 and 2? #1 negative, #2 positive


x

y

x

y

162

No Relationship - scattered

Example: Example: hair color, grades

Can someone give me another example of a scattered relationship?

You Try! Determine whether a scatter plot of the data for the following might show a positive, negative, or no

relationship.

1. Time spent in the gym and your strength. Positive

2. The amount of songs on your iPod and the amount of space left. Negative

3. Total text messages and your bill. No relationship (if you have unlimited plan)

Definition Best fit line – line that summarizes the data set

Things to remember:

1. Follow the basic direction of the data

2. Same amount of points above and below the line

3. Draw line through as many points as possible

Example 1: Draw a best fit line

x

y

Time Studying vs. Test Score

60

65

70

75

80

85

90

95

100

0 1 2 3 4

Time studying (Hours)

Tes

t S

core

(P

erce

nt)

Test score

163

Example 2: Draw a best fit line

Time at the Mall vs. Money Remaining

0

50

100

150

200

250

0 2 4 6 8 10

Time (Hours)

Mo

ney (

$)

You Try! Draw a best fit line for each of your two homework problems.


1. Determine whether a scatter plot of the data for the following might show a positive, negative, or no

relationship.

a. Age of a car and value of the car.

b. The size of a family and the weekly grocery bill.

c. The size of a car and the cost.

d. A person’s weight and percent body fat.

e. Time spent playing video games and time spent on outdoor activity.

2. Draw a best fit line for each graph.

a.


164

b.

3. The following data set represents the average salary (in thousands) for people who have a four year

college degree.

Year 1999 2000 2001 2002 2003 2004 2005 2006

Salary

(Thousands) 52 55 58 62 62 66 67 69

a. Draw a scatter plot based on the data set.

b. Draw a best fit line.

c. What type of relationship exists between the two sets of data?

d. Predict the average salary in 2010.

e. Predict the year in which salaries will be $100,000.

4. The following data set represents the miles traveled and how much gas is left.



c. What type of relationship exists?

d. How many gallons should be left after you travel 300 miles?

e. How far did you travel if you have 12 gallons left?

f. What factors cause the points not to be in a straight line?

Miles Traveled 25 100 110 140 220

Gallons of gas 14 10 8 7 4

165

Review Question Explain how to draw a best fit line.

1. Follow the basic direction of the data

2. Same amount of points above and below the line

3. Draw line through as many points as possible

Discussion What two things do you need to write an equation of a line? Slope, y-intercept

What would the equation of this best fit line be?

Time at the Mall vs. Money Remaining

0

50

100

150

200

250

0 2 4 6 8 10

Time (Hours)

Mo

ney

($)

y = __x + __

The y-intercept is 225. Then calculate the slope by using two points on the best fit line.

(1, 200) and (7, 90)

m = 110/-6

y = 110/-6 x + 225

SWBAT write the equation of the best fit line

Example 1: Days and money saved. Draw a scatter plot and best fit line. Then write the equation of the line.

DAYS Money

Saved

2 20

12 35

17 38

25 65

35 75

40 80


166

(0, 15) (35, 75); m = 60/35

y = 60/35x + 15

You Try! Write the equation of the best fit line for problems 3 and 4 from last night’s homework.


1. Determine whether a scatter plot of the data for the following might show a positive, negative, or no

relationship.

a. The height of a person and their shoe size.

b. The amount a student talks in class and their grade.

c. The amount of shots a basketball player takes and the amount of shots they make.

d. The color of someone’s shoes and their grade.

2. Write the equation of the best fit line.

a.


167

b.

3. The following data set represents the grade a person is in and their IQ.

Grade 5 6 7 8 9 10 11 12

IQ 75 78 85 100 102 120 125 140



c. Write the equation of the best fit line.

d. What type of relationship exists between the two sets of data?

e. Predict the IQ after two years of college.

4. The following data set represents the amount of songs on your IPod and the amount of space left.



c. Write the equation of the best fit line.

d. What type of relationship exists between the two sets of data?

e. Predict the amount of space that would be left with 1200 songs.

f. What factors would cause the point not to be in a straight line?

Songs 100 250 300 400 650 750 1000

Space Left

(Gigs) 7.5 7.1 6.7 6.3 5.4 5.1 4.3

168

Review Question How do you write the equation of the best fit line?

Use y = mx + b. Find the y-intercept (b). Then locate two points on the line. Then find the slope

(m) between the two points.

Discussion Scatter plots and best fit lines are used in engineering. When engineers are designing roadways they must

calculate how many lanes of traffic and traffic lights are needed. In order to do this, they collect data.

They collect data on how many cars are added to the roads for different size housing plans. This data is

then graphed on a scatter plot.

Once the data is graphed, a best fit line and equation are developed. The engineers use this equation the

next time someone wants to put in a housing plan. They enter the amount of new homes into the best fit

equation and get a value for how many new cars the development will add to the current roadways.

The engineers use this information to figure out how many new lanes and lights will be needed.

SWBAT will make up a data set that represents a positive/negative relationship

Activity If you truly understand something, then you can talk freely about it. Specifically, you should be able to

come up with your own explanations about the topic. This is what we will be doing today.

1. Make up a data set (at least 10 points) that has a positive relationship. Then do the following:

a. Write a sentence describing your data set.

b. List your data set in a table.

c. Make a scatter plot.

d. Draw a best fit line.

e. Find the equation of the best fit line.

2. Make up a data set (at least 10 points) that has a negative relationship. Then do the following:

a. Write a sentence describing your data set.

b. List your data set in a table.

c. Make a scatter plot.

d. Draw a best fit line.

e. Find the equation of the best fit line.



169

Review Question What does a graph of a positive relationship look like? Up/right

What does a graph of a negative relationship look like? Down/right

SWBAT study for the Unit 4 Test

Discussion 1. How do you study for a test? The students either flip through their notebooks at home or do not

study at all. So today we are going to study in class.

2. How should you study for a test? The students should start by listing the topics.

3. What topics are on the test? List them on the board

- Slope

- Slope Intercept Equation

- Point Slope Equation

- Parallel and Perpendicular

- Scatter Plots

4. How could you study these topics? Do practice problems

Practice Problems

Have the students do the following problems. They can do them on the dry erase boards or as an

assignment. Have students place dry erase boards on the chalk trough. Have one of the groups explain

their solution.

For problems 1-6, find the slope, y-intercept, x-intercept, and graph.

1. Write an equation of a line in slope intercept form that passes through (5, 8) and (4, 3).

y = 5x – 7, m = 5, y-int = -7, x-int = 7/5

2. Write an equation of a line in slope intercept form that has a slope of -3 and passes through

(-4, 3). y = -3x – 9, m = -3, y-int = -9, x-int = -3

3. Write an equation of a line slope intercept form that is parallel to y = 3x – 7 and passes through

(7, -2). y = 3x – 23, m = 3, y-int = -23, x-int = 23/3

4. Write an equation of a line in slope intercept form that is perpendicular to y + 4x = 5 and passes

through (-3, -3). y = 1/4x – 2 1/4, m = 1/4, y-int = 2 1/4, x-int = 9

5. Write an equation of a line in slope intercept form that is parallel to x = 3 and passes through (2, 6).

x = 2, m = undefined, y-int = none, x-int = 2

6. Write an equation of a line in slope intercept form that is perpendicular to y-axis and passes through

the origin. y = 0, m = 0, y-int = 0, x-int = All Reals

Unit 4 Review

170

7. Given the following data sets:

a. Draw a scatter plot.

b. Does a positive or negative relationship exist? +

c. Write the equation of the best fit line. y = 1.5x + 10

d. How many people should the cinemas expect if there are 345 cars in the lot? 600

e. What factors cause this data not to be perfectly linear? (Hint: my mom can drive

one way to the movies) Different amounts of people/car.

f. How could this information help the owner of the cinema? Less cars = more people riding

together = more families = more family movies

# of cars in movie

cinemas lot

# of people in the

movie theatre

75 135

100 168

125 220

150 305

175 315

200 425

171

172

SWBAT do a cumulative review

Discussion What does cumulative mean?

All of the material up to this point.

Does anyone remember what the first four chapters were about? Let’s figure it out together.

1. Pre-Algebra

2. Solving Linear Equations

3. Functions

4. Linear Equations

Things to Remember:

1. Reinforce test taking strategies: guess/check, eliminate possibilities, work backwards, and estimating.

2. Reinforce the importance of retaining information from previous units.

3. Reinforce connections being made among units.

1. What set of numbers does 5

1belong?

a. Counting b. Whole c. Integers d. Rationals

2. 2(4x + 3) = 8x + 6 is an example of what property?

a. Commutative b. Associative c. Distributive d. Identity

3. What is the value of -5 + 12 ?

a. 17 b. -7 c. 7 d. -11

4. What is the value of -8 – 12 ?

a. -20 b. 20 c. 4 d. -4

5. What is the value of 12 – 8.2 ?

a. 20.2 b. 3.8 c. 4.8 d. 21

6. What is the value of (-2.5)(.34) ?

a. -1.7 b. -85 c. -8.5 d. -.85

7. What is the value of -1.488 ÷ .24 ?

a. -6.2 b. -62 c. -.062 d. -6

8. What is the value of 9

2

6

12 ?

a. 17/18 b. 16/18 c. 43/18 d. 35/18

In-Class Assignment

UNIT 4 CUMULATIVE REVIEW

173

9. What is the value of 4

3

2

19 ?

a. 35/3 b. 38/3 c. 1/2 d. 12/3

10. 33

a. 3 b. 6 c. 9 d. 27

11. 441=

a. 21 b. 29 c. 220.5 d. 87

12. 32 =

a. 16 b. 2 4 c. 6 d. 4 2

13. 103 ÷ (-2 • 5) – 2

a. -96 b. -98 c. -100 d. -102

14. 10(2x) – 2(4x) =

a. 12x b. -28x c. 28x d. 14x

15. (x + 5) – (3x + 10) =

a. x + 10 b. 4x + 6 c. -2x – 5 d. -2x + 15

16. 4(2x + 12) + 4 = 3x + 52 + 5x

a. -25/3 b. 22/5 c. Empty Set d. Reals

17. 435

x

a. 35 b. -35 c. 5 d. 1

18. -3(2x + 7) = -6x + 11

a. 2 b. 1 c. Empty Set d. Reals

19. 28

5

x

a. 21 b. -11 c. 20 d. 14

20. Solve for y: -2(3a + y) = -5b

a. ab

y 32

5 b. y = 5b – 2a c.

2

5by d.

23

yay

21. Solve y = -x – 2; given a domain of {-2, 0, 5}

a. -2, 4, 5 b. 0, -2, -7 c. 0, 2, 5 d. 1, 2, 3

174

22. y = 2x – 4

a. b. c. d.

23. y = 3

a. b. c. d.

24. If f(x) = 4x – 5, find f(3).

a. 7 b. 8 c. 9 d. 10

25. Which equation is not a linear equation?

a. 724 yx b. yx

4 c. 5x d. 32 xy

26. Which equation is not a function?

a. 73 xy b. 5y c. 5x d. 22

1 xy

27. Write an equation for the following relation: (2, 10) (6, 8) (10, 6).

a. xy 2 b. 124 xy c. 112

1 xy d. 112 xy

28. Write an equation of a line that passes through the points (2, 10) and (7, 20).

a. y = -2x + 6 b. y = 4x + 12 c. y = 2x + 6 d. y = 2x – 11

29. Write an equation of a line that is perpendicular to 33

1 xy and passes through the point (-2, 4).

a. y = -3x + 6 b. y = -3x – 2 c. y = 3x + 6 d. y = 2x – 11

30. Write an equation of a line that is parallel to y + 4x = -5 and passes through the point (5, -3).

a. y = -4x + 17 b. y = 4x + 12 c. y = -4x + 6 d. y = 2x – 11

31. Write an equation of a line that is parallel to the y-axis and passes through the point (-2, -5).

a. x = -4 b. x = -2 c. y = -4 d. y = -2

32. Write an equation of a line that passes through the point (-2, 4) and has a m = 3.

a. y = 3x – 2 b. y = 3x – 10 c. y = 3x + 10 d. y = 3x

33. Write an equation of a line that has m = -2 and a y-intercept of -3.

a. y = 2x – 3 b. y = -2x – 10 c. y = -2x – 3 d. y = -2x

175

34. What is the x-intercept of the line y = 3x + 9?

a. 9 b. 3 c. -3 d. 0

35. Which of the following equations could be the best fit line for sets of data that include time traveled

and miles traveled?

a. y = 45x b. y = -45x c. y = 45x + 25 d. y = 45

176

1. What are the slope and y-intercept represented in the graph?

a. m = -1, b = -2 b. m = 1, b = -2 c. m = 1, b = 2 d. m = -1, b = 2

2. What is an equation for the line that passes through the coordinates (2, 0) and (0, 3)?

a. y = -3/2x + 3 b. y = -2/3x + 2 c. y = -3/2x – 3 d. y = -2/3x – 2

3. Which of the following is the equation of a line with a slope of 0 and passes through the point (4, 6)?

a. x = 4 b. x = -4 c. y = 6 d. y = -6

4. What are the slope and y-intercept of the line 7x – 3y = 4?

a. m = 7/3, b = -4/3 b. m = -7/3, b = 4/3 c. m = -7/3, b = -4/3 d. m = 7/3, b = 4/3

5. Johnny’s restaurant sells hamburgers. The amount charged for a hamburger (h) is based on the cost

for a plain hamburger plus an additional charge for each topping (t) as shown in the equation: h = .60t + 5.

What does the number .60 represent in the equation?

a. The number of toppings

b. The cost of a plain hamburger

c. The additional cost for each topping

d. The cost of a hamburger with one topping

Standardized Test Review

177

6. The following problem requires a detailed explanation of the solution. This should include all

calculations and explanations.

Joey collected data on how long he studies and what his test grade is.

a. Based on these two data points, write an equation of a line in slope intercept form.

Minutes

Studying Grade

0 55

5 65

b. What is the slope, y-intercept, and x-intercept of the line?

c. Based on the following data set, draw a scatter plot and best fit line.

Minutes

Studying Grade

0 55

5 65

10 72

15 70

20 85

25 98

d. Write an equation for the best fit line.

e. What type of relationship exists between the two sets of data?

Documents

Unit 4 Analyzing Linear Equations - Aliquippa School … Manual42.pdf131 Unit 4 – Analyzing Linear Equations 4–1 Slope 4–2 Slope Intercept Equation 4–3 Point Slope Equation