the adventures of bob: analyzing and graphing linear equations

The Adventures of Bob: Analyzing and Graphing Linear Equations

Help Bob through his challenges in medieval times, space, the rainforest and the financial

jungle by graphing and analyzing linear equations.

EDTEC 544—Cynthia Jackson and Sonya Rea

This is Bob. He is about to find a portal in time and have the adventure of a lifetime. He will need your help. Are you ready?

Each time you master a skill you will see this space

portal.

Click to Open Bob’s Notebook

As you go through each level of the game you will learn about each skill you will need to help Bob. If you need help, click on Bob’s notebook to look at the

notes for that level.

Level One—Medieval Kingdom

Bob skates his way into medieval times. You can help him become king by collecting swords, shields, and slaying a dragon. Good luck!

Information You Will Need to Help Bob in Level One

Slope: The slope of a line measures the steepness of the line. Most of you are probably familiar with associating slope with "rise over run". Rise means how many units you move up or down from point to point. On the graph that would be a change in the y values. Run means how far left or right you move from point to point. On the graph, that would mean a change of x values.

Here is a visual to help you with this definition:

Positive slope: Note that when a line has a positive slope it goes up left to right.

Negative slope: Note that when a line has a negative slope it goes down left to right.

Bob’s NotebookLevel One—Medieval Kingdom

Bob’s NotebookLevel One—Medieval Kingdom

More Information You Will Need to Help Bob in Level One

I am ready to help Bob!

y-intercept: The y-intercept of a line is the point at which a line crosses the y-axis.

Here is a visual to help you with this concept:

How To find the y-intercept:Step 1: Put your finger on the origin (0,0).Step 2: Now find your line. Is it above your finger or below your finger? If it is above your number will be positive, if it is below your number will be negative. Step 3: Now move your finger up or down until it touches the point where your line crosses the y-axis. Where did you end up? This is the y-intercept.

“If the line is going

down it is negative!”

Bob needs your help. Identify whether the slope of the line is positive or negative. If you do it correctly Bob will skate up or down the ramp.

Positive

Negative

Next Problem

x

y


“If the line is going up

it is positive!”

Bob needs your help. Identify whether the slope of the line is positive or negative. If you do it correctly Bob will skate up the ramp.

Positive

Negative

Next Problem


“Try again!

Bob needs your help. Identify whether the slope of the line is positive or negative. If you do it correctly Bob will skate down the ramp. Click on the space portal after Bob

skates.

Positive

Negative


You have helped Bob so well that he is skating better than ever! He has found a space portal. Click on Bob

to help him skate into the time portal and back into medieval times.


“Find the

origin!”“Find the y-axis!”

“Find the x- axis!”

Bob needs your help. Identify the coordinates of the y-intercept of the given line. If you do it correctly Bob will collect swords and shields that will help him slay a dragon.

(0, 0) (-1, -1)

(-2, 2) (-3, -3)

Next Problem

y

x

“x is written

before y!”

“This is the x-

intercept!”


(-2, 2) (2, 0)

(-2, 0) (0, 2)

Next Problem

“Look at where the line crosses

the y axis!”


“Check my Notebook!

”

“This is the x-

intercept!”


(-2, 0) (0, -2)

(-4, 0) (0,- 4)

Next Problem

“Check my notebook!”



“x is always written first!”

“This is the x-

intercept!”

“Look at where the

line crosses the y-axis!”

Good job! Now Bob needs a shield. Identify the coordinates of the y-intercept of the given line. Help Bob to collect a

shield so he can slay the dragon.

(-3, 0) (2, -2)

(2, 0) (0, -3 )

Next Problem

Hint: the slope is negative.

Hint: the slope is negative.

Bob needs your help to slay the dragon and become the king. Find the equation of the line based on the y-intercept and the slope.

y = - x + 2

y = 2x + 2

y = x + 2

y = - 1x + 3

Hint: the y-intercept is 2.

Click here after you help Bob slay the dragon

Click on me


Level One AssessmentHelp Bob pass this test with 80% accuracy so he can move on to Level Two.

1. In Coordinate Plane 1, is the line positive or negative?

1

Positive Negative

Next Problem

(-1, 0) (1, 0)

(-2, 0) (0, -2)

2. What are the coordinates of the y-intercept of line A ?

A

3. What are the coordinates of the y-intercept of line A ?

(0, 3)

(4, 0) (3, 0)

(0, 4)

4. Find the equation of the line based on the y-intercept and the slope.

y= 2x + 3

-3y= x - 3y= x 3

y= -3x - 3

Next Problem

Check Your Results

- y = x - 4

y = 4x - 4y = x + 4

y = -2x + 2

5. Find the equation of the line based on the y-intercept and the slope.

You answered …

4Out of five questions correctly. This

gives you a percentage of…

80% Congratulations on Passing Level One!

Congratulations! You have mastered Level One. Click on Bob to travel into the time portal.

The x-coordinate is always the first number in the pair and the y-coordinate is always the second number in the pair. The point below shows the following ordered pair:(4, -2). The point is represented by a black dot. Therefore (x,y) = (4, -2).

To identify the ordered pairs or to locate points, you start at the origin and count the units along each axis. This point show 4 to the right, and 2 down.

Information You Will Need to Help Bob in Level Two

Bob’s NotebookLevel Two—Space

Now that Bob is King he wants to rule the universe. He needs your help to launch his spaceship and cross the evil empire of Yint.


Steps for Slope FormulaFormula for Slope:

In order to use the slope formula, you need to be given 2 points.Example, (4,3) and (5,-2)

1. Steps:2. Label your points with the x and y labels3. Substitute your labeled values into the formula4. Simplify both the numerator and the denominator

2 35 4

2 35 4

51

EX: Find the slope of a line with the points (4,3) and (5,-2)Step 1-Label your points with the x and y labels (4,3) and (5,-2)

x1 y1 x2 y2Step 2-Substitute your labeled values into the formula

Slope= -5

The slope intercept form of a line is y = mx + b.

Where m is the slope and b is the y-intercept.

The y-intercept is the point where the line crosses the y-axis.

In the example to the right, the line crosses the y-axis at point (0,3).

Therefore b = 3.

The slope is 2.

The slope intercept equation isy = 2x + 3.


Steps for finding coordinates of y-intercept based on 2 points.

1.Find the slope of the line based upon the two points

2.Using the slope, count coordinates and fill in points on the graph

3.Label the coordinate that crosses the Y-axis. That is your y-intercept

Example: Points (2,-2) and (-2,2) lie on the same line. Find the

coordinates of the y-intercept of that line.

Step 1-Find the slope of the line


Slope= 2 ( 2) 2 2

2 2 4

4 4

11

Step 2-Using the slope, count the coordinates

So, starting with point (2,-2), you would rise -1 and run +1

Your next coordinate would be (1,-1) and then (0,0).

Step 3-Label the coordinates

Because your line passes through point (0,0), your y-intercept is (0,0)

Finding an equation of a line based on 2 points:

1. Find the slope of the line with the 2 given points

2. Using the slope-intercept form of y=mx+b, write your equation.

Coordinate Pair (1,-2) and (5,-6) lie on the same line. Give the equation of that

line.

Step 1-Find the slope of the line

3. Find the y-intercept of the line by graphing



6 ( 2)5 1

6 24

44

1

Step 3-Use the slope-intercept formy=mx+by=-1x+(-1), so your equation is…y=-x-1

(4, 3)

(5, -2)

Use the Slope Formula to help Bob find the right

slope to launch his spaceship.

5

4

1/3

-5

Check your sign.

The y-coordinates

are the numerator. Divide.

Don’t subtract.

Next Problem

x

y

rise

run=

Way to go! One more right

answer and I can launch the

spaceship. Click to Open Bob’s Notebook

(4, 1)

(1, -2)

Use the Point Slope Formula to help Bob find the right slope to launch

his spaceship.

-1

0

1/2

1

Check your sign.

Try again.

Divide. Don’t

subtract.

Next Problem


You need to help Bob cross the virtual space fence of Yint, the Evil Empire.

Points ( 2, -2) and ( -2, 2) lie on the same line. Find the coordinates of

the y-intercept of that line.

YINT(-1, 0)

(2, -2)

(0, 0)

(-2, 2

“This is one of the coordinates.”

“ This is one of the

coordinates.”

“Check Bob’s

Notebook.”

Next Problem

x

y


(-2, 2)

(2, -2)

“ One more correct answer and we can

cross the space fence of YINT!.”

You need to help Bob cross the virtual space fence of Yint, the Evil Empire.

Points ( 1, -2) and ( -3, 2) lie on the same line. Find the coordinates of

the y-intercept of that line.

YINT(-1, 0)

(-4, 4) (0, -1)

(0, 1)

“x always goes before y!”

“Keep working. Divide these

numbers to find the slope!”

“Check your

sign!”

Next Problem


Coordinate pair (1, - 2) and (5,- 6) lie on the same line. Give the equation of the line.

You and Bob have mastered slope and y-intercept. Using the two coordinate pairs find the equation of the line so that Bob can become the ultimate Space Tyrant, ruler of all four quadrants of space.

II am the ultimate

Space Tyrant,

ruler of all four

quadrants!

y = x + 1

y = -x - 1

y = x - 3

y = 2x -1

“The y-intercept is negative!”

“The slope is negative!”

“The slope is not 2. Use the point intercept

formula!”


Congratulations! You have helped Bob become the Ultimate Space Tyrant.

Now you must complete the assessment to determine Bob’s preparedness for his next adventure.

1. A given line contains the points (1,1) and (2,-1). What is the slope of that line?

y=1/2 x – 3/2 y= 2x-4 y=-2x-3

y= 1/2x -4y= ½ x -3

(0,-1)(0,1)

(0, -3)(0,-4)1/2-2

-32

y= 1/2x + 2

y= -1/2 x +2 y= -1/2x – 1/2

2. A line contains the points (-2,-2) and (1,-5). What is the y-intercept of that line?

3. A line contains the points (3,0) and (-1,-2). What is the equation of that line?

4. A line connects points (-3,3) and (1,-5). What is the equation of the line?

Next Problem

5. The line connecting Bob’s planet with the nearest time portal contains points (-1,-1) and (0,1). Bob needs to know the equation of that line in order to navigate to the time portal. What is the equation of the line?

y= -2x-1y= -1/2x -1

y=2x+1y= 1/2x + 1

Check Your Results

You answered …



100% Congratulations on Passing Level Two!

Congratulations! You have mastered Level Two and helped Bob become the Ultimate Space Tyrant. Click on

Bob to travel into the time portal.

Parallel Lines As you can see from the

diagram below, these lines have the same slope and are

never going to intersect:

Perpendicular Lines As you can see from the picture below: the

slope of these lines are negative reciprocals: (Ex: 2 and - ½ are negative reciprocals)

these lines are perpendicular and intersect at 90 degrees

Bob’s NotebookLevel Three—The Rainforest

Now that Bob is the Ultimate Space Tyrant he wants to go back to Earth and find treasure that has been hidden by evil gorilla poachers. He needs

your help to make his way through the jungle.

Parallel Perpendicular Dependent

Slope Slope is the same

Slope is thenegative reciprocal

When factored, slope is the same

Equation Slope is the same

Intercept is the same

When factored, equation is the same.


Steps for finding the equation of a line given a parallel line and a set of points

1. Look at the equations of your two lines:Steps for finding a linear equation in slope-intercept form:1.Use the formula: y-y1=m(x-x1) x1 and y1 is the point you were given2.Substitute the point into the formula3.Simplify by isolating and solving for yWhat is the equation of line A given the point (2,2) and is parallel to line B y=3x-1?Remember, parallel lines contain the same slope, so your slope is 3Step 1 -Use the formula:y-y1=m(x-x1)Step 2 -Substitute the given point into the slope-intercept formy-2=3(x-2)Step 3 -Simplify by solving for yy-2=3(x-2)y-2=3x-6 +2 +2y =3x-4ANSWER: y=3x-4



Line A is y = 2x + 4. Line B is 3y = 6x + 12. What do you know about these lines?

Bob needs to build equivalent poles in order to make his way over the snake pit.

The lines intersect

The lines are perpendicular

The lines have different slopes

The lines are dependent

“These lines can’t intersect because they

have the same slope.”

“Perpendicular lines have a reciprocal slope

and it differs in sign.”

“Try dividing both sides of line B by

3.”

Next Problem


Thanks for helping me! Click on me so I can

make my way across the snake pit.”

Line A is -2y = 6x - 4. Line B is y = -3x + 2. What do you know about these lines?

The lines intersect at

(0,0)

The lines have a positive

slope

The lines are perpendicular

The lines are dependent

“These lines can’t intersect because they

have the same slope.”

“Perpendicular lines have a reciprocal slope

and it differs in sign.”

“These lines have a negative slope.”

Next Problem


Bob needs to unlock the code to the hidden bridge so he can cross the gorge. Can you help him?

These two lines are parallel. This means that their ________ are equal.

origins

slopes

y-intercepts

x-intercepts

“Look at where each line on the graph

crosses the y-axis.”

“Look at where each line on the graph

crosses the x-axis.”

“Only one of these lines goes through

the origin.”

Next Problem


“Great job! One more right answer and we can unlock

the code and cross the bridge. Click Next

Problem.”

Bob needs to unlock the code to the hidden bridge so he can cross the gorge. Can you help him?

Line A is y = 2x – 3.Line B is y = 2x + 5. What do you know about these lines?

These lines are perpendicular

The lines are parallel

They will intersect

They are the same line

“These lines will never intersect.”

“If they were the same line, besides having the same slope they would have the same y-

intercept.”

“Perpendicular lines have negative

reciprocals.”

When the bridge

appears click on Bob so he

can cross over.

Next Problem


Help Bob rescue the

hidden treasure. It is being guarded by endangered gorillas. Bob needs to be

careful not to hurt them.

Answer carefully. Don’t

alert the poachers to

Bob!

Line A contains the point (2, 2) and is parallel to line B y = 3x - 1. What is the equation of line A?

y=-3x + 1

y = 3x + 4

3y= 3x + 4

y= x - 2

“Be careful! They’re shooting at us. The line you

chose has the same y-intercept not the same

slope.”

“Be careful! They’re shooting at us. The line

you chose has a negative slope. “

“Be careful! They’re

shooting at us. Check Bob’s

notebook.”

Click on Bob when if he is able to retrieve the

treasure


Level Three AssessmentHelp Bob pass this test with 80% accuracy so he can move on to Level Four.

1. Line A is y = -x + 3 Line B is -4y = 4x – 12.

What can you say about these lines?

These are dependent

lines

The lines are parallel

The lines intersect at

(0,0)

They are perpendicular

lines

Next Problem

2. These two lines are parallel. This means that they have ______ that are equal.

Next Problem

x-intercepts

y-intercepts

origins

slopes

These are perpendicular

lines

The lines will intersect at the

y-intersect

These are parallel lines

They are the same line

3. Line A is y = -2x + 3 Line B is y = -2x – 5.

What can you say about these lines?

4. Line A contains the point (1, 2) and is parallel to line B y = 4x + 3. What is the equation of line A?

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

y = 4x y = x + 2

Next Problem

5. Line A contains the point (1, 6) and is parallel to line B y = x - 2. What is the equation of line A?

y = x + 6

y = x + 5y = x + 2

y = 6x + 1

Check Your Results

You answered …



100% Congratulations on

Passing Level Three!

Congratulations! You have helped

Bob find the treasure and master Level Two. Click on him so he can

run through the time portal.

Bob’s NotebookLevel Three—Financial World

1 12

y x

12

Equation of line A is

Remember, perpendicular lines have slopes that are negative

reciprocals to each other.

Therefore, the slope of Line B would be -2, because the slope of line

A is .

.

Line B is perpendicular to line A. What is the slope?

Perpendicular Lines: Two lines are considered perpendicular to each other if they intersect to form right angles.

Parallel Perpendicular DependentSlope Slope is the

sameSlope is the negative reciprocal

When factored, slope is the same

Equation Slope is the same

Intercept is the same

When factored, equation is the same.

Bob’s NotebookLevel Three—The Financial World

Steps to finding an equation of a perpendicular line given a point

1. Find the slope of the given line

2. Find the reciprocal of the slope of the given line3. Use the slope-intercept form equation y-y1=m(x-x1) to find the new line

Line A contains (-2,4) and is perpendicular to line B y=2x+1. What is the equation of line A?Step 1 - Find the slope of the given line. y=2x+1, so the slope of this line is 2.

Example

Step 2-Find the reciprocal of the slope of the given line.If the slope of our given line is 2, then our reciprocal is -

-

12

.

Step 3-Use the slope-intercept form equation. y-y1=m(x-x1)y-4=-1/2(x-(-2))

y 4 12(x 2) y 4

12x 1

y 12

3

+4 +4

______________

ANSWER:

Bob needs money to compete in the financial world. Help him answer correctly and add money to his bank account.


These two lines are perpendicular. That means that they intersect to form ________ angles.

Next Problem

“Click on my notebook for

help.”

“Click on my notebook for

help.”

“Perpendicular lines form right angles.”

negative

positive obtuse

right

Now that Bob has money, he plans to invest it in real estate. Help Bob acquire property by finding the slope of the line which is perpendicular to the given line. If you select the correct answer, he makes a wise purchase. If not,…

Bob needs to know the slope of a line which is perpendicular to y=-1/3x + 1. The slope of that line would be..

-1/3

1/3

3

-3


With your help, Bob can conquer the real estate world. You will be given a point and the equation of a line perpendicular to the one Bob needs. Give the equation of the line Bob needs, so he can get from home to work and make the deal of a lifetime!

The line from home to work contains point (2,1) and is perpendicular to y=-2x+3. What is the equation of the line?

y=-2x+0

y=-1/2x+3

y=1/2x+0

y=2x+3


“”Look in my notebook for

help.”

“”Check your sign.”

“”Look in my notebook for

help.”

Level Four AssessmentHelp Bob pass this test with 80% accuracy so he can go to his next adventure in Algebra.

1. Perpendicular lines create what angle?

A right angle

A left angle A 45 degree angle

A straight angle

2. If a given line has a slope of 2, the line perpendicular to that one will have a slope of

2

-1/21/2

-2

3. If a given line contains the point (3,1) and is perpendicular to line y=4x-2, what is the equation of the given line?

y= -1/4x – 7/4 y=-1/4x + 7/4

y=-4x - 7/4y=1/4x - 1/4

4. Bob’s line is perpendicular to Hank’s line. Bob’s line has the equation y=-4x+2. Hank’s line contains the point (0,0). What is the equation of Hank’s line?

y = ¼ x

y= -1/4 x + 2

y = ¼ x + 2

y= -1/4 x

You are almost finished! Just a few more questions remain.

5. Bob and Sally are planning to have a road built to connect Bob’s street with Sally’s home. Because Bob likes right angles, the new road will, of course, be perpendicular to Bob’s street. Bob’s road follows the equation y=-1x +1. Sally’s home is at point (5,5). What will be the equation of the line for the new road?

y=x - 1 y= -x - 4

y=x y=1/4x

Check Your Results

You answered …



80% Congratulations on Passing Level Four!

Congratulations! You and Bob have mastered

Graphing and Analyzing Line Equations! Join Bob on his next adventure in Algebra: Polynomials.

Click on Bob to send him and Sally through the

Time Portal.

You answered …



60% Please return to Bob’s Notebook

and review Bob’s notes

the adventures of bob: analyzing and graphing linear equations

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