Algebra Period 5)

Chapter 4-4 Parallel and Perpendicular Lines

Parallel Lines:

2 lines that are parallel to each other have the SAME SLOPE

Example:

y = 2x – 10

y = 2x + 4

Are parallel because they both have a slope of 2

BE CAREFUL:
to be parallel they must have the SAME SLOPE BUT DIFFERENCE y-INTERCEPTS

If they have the same m and the same b then they are COLLINEAR ( or they COINCIDE) they are the same line and NOT parallel

Perpendicular Lines

2 lines that are perpendicular to each other have SLOPES that are:

Opposite signs
and
Reciprocals

y = 2x – 10 is perpendicular to y = (-1/2)x + 4

This is another twist to our Mystery Line Puzzles…

If you know that the mystery line is parallel  or perpendicular to another given line they you know the mystery line’s slope!!

EXAMPLE 1:

 Your mystery line has a point of (2, -5) and is PARALLEL to the line y = 2x + 4

So you know that the mystery line’s slope
because it is the same as the given line  (m = 2)

Substitute the slope and the point given for the mystery line and solve for b OR use Point Slope Form and solve easily!
This is why I like Point Slope: y + 5 = 2(x – 2)

y + 5 = 2x – 4

y = 2x – 9

If I used Slope Intercept form or y = mx + b

-5 = 2(2) + b

-5 = 4 + b

b = -9

Plug everything back in

y = 2x – 9

Same results—but I think that it takes longer (students also tend to FORGET to put everything together)

EXAMPLE 2:

Your mystery line has a point of (2, -5) and is perpendicular to the line y = 2x + 4

So you know that the mystery line’s slope is the opposite sign and the reciprocal of the given line
(since the given line’s slope is 2, the mystery line’s slope must be –½ )

Substitute the slope and the point given on the mystery line and solve for b OR use Point Slope Form and solve easily!  

y + 5 = (-½)(x -2)

y + 5 =  -½x + 1

y =  -½x - 4

If I used Slope Intercept form or y = mx + b

-5 = (-½)(2) + b

-5 = -1 + b

-4 = b

Plug everything back

 y = -½x - 4 

Algebra Honors (Period 4)

Chapter 4-4 Parallel and Perpendicular Lines

Parallel Lines:

2 lines that are parallel to each other have the SAME SLOPE

Example:

y = 2x – 10

y = 2x + 4

Those two lines are  parallel because they both have a slope of 2

BE CAREFUL:
To be parallel they must have the SAME SLOPE BUT DIFFERENCE y-INTERCEPTS

If they have the same m and the same b then they are COLLINEAR
(or they COINCIDE) they are the same line and NOT parallel

Perpendicular Lines

2 lines that are perpendicular to each other have SLOPES that are:

Opposite signs

and

Reciprocals

y = 2x – 10 is perpendicular to y = (-1/2)x + 4

This is another twist to our Mystery Line Puzzles…

If you know that the mystery line is parallel  or perpendicular to another given line they you know the mystery line’s slope!!

EXAMPLE 1:

Your mystery line has a point of ( 2, -5) and is PARALLEL to the line y = 2x + 4

So you know that the mystery line’s slope
because it is the same as the given line  (m = 2)

Substitute the slope and the point given for the mystery line and solve for b
OR use Point Slope Form and solve easily!
This is why I like Point Slope: y + 5 = 2(x – 2)

y + 5 = 2x – 4

y = 2x – 9

If I used Slope Intercept form or y = mx + b

-5 = 2(2) + b

-5 = 4 + b

b = -9

Plug everything back in

y = 2x – 9

Same results—but I think that it takes longer ( students also tend to FORGET to put everything together)

EXAMPLE 2:

Your mystery line has a point of ( 2, -5) and is perpendicular to the line y = 2x + 4

So you know that the mystery line’s slope is the opposite sign and the reciprocal of the given line
(since the given line’s slope is 2, the mystery line’s slope must be –½ )

Substitute the slope and the point given on the mystery line and solve for b OR use Point Slope Form and solve easily!  

y + 5 = (-½)(x -2)

y + 5 =  -½x + 1

y =  -½x - 4

If I used Slope Intercept form or y = mx + b

-5 = (-½)(2) + b

-5 = -1 + b

-4 = b

Plug everything back

 y = -½x - 4

Algebra (Period 5)

Chapter 4-3 Point Slope Form

There is another way to solve for a line other than using slope-intercept form  Although most people use the slope- intercept form for all cases, the POINT-SLOPE Form  is actually easier—everything is built into the format!  You do not need to “put it altogether” at the end!

MY FAVORITE!!

It gives you exactly a point and the slope—just by looking at it!

You do not need to do anything BUT graph!!

Point- Slope Form of a line:

y-y₁ = m ( x- x₁)

Example: ( 3, 1) is a point on the line and m = 2

y – 1 = 2( x – 3)

What you have now is point-slope form of the line!

You can graph easily starting at (3, 1) and going up 2 and to the right 1

You can also simplify this and get the slope- intercept form of the line:

y - 1 = 2x – 6

y = 2x – 5

If you are trying to link the Slope-Intercept form to the Point-Slope form of the line:  The point slope version eliminates one step from using the slope intercept form

In the slope intercept form, you plug in the point and the slope and solve for b—
AND THEN rewrite the equation using the intercept that you found.

In point slope form, once you plug in the point and slope, you just simplify and the equation is already done!

IF you don’t have the slope, you will need to compute it with the formula—just like you did with Slope Intercept FORM

The biggest difference is that any point is plugged into this form, while  the Slope- Intercept Form focuses on the ONE specific point on the y axis 

Algebra Honors (Period 4)

Chapter 4-3 Point Slope Form

There is another way to solve for a line other than using slope-intercept form  Although most people use the slope- intercept form for all cases, the POINT-SLOPE Form  is actually easier—everything is built into the format!  You do not need to “put it altogether” at the end!

MY FAVORITE!!

It gives you exactly a point and the slope—just by looking at it!

You do not need to do anything BUT graph!!

Point- Slope Form of a line:

y-y₁ = m ( x- x₁)

Example: ( 3, 1) is a point on the line and m = 2

y – 1 = 2( x – 3)

What you have now is point-slope form of the line!

You can graph easily starting at (3, 1) and going up 2 and to the right 1

You can also simplify this and get the slope- intercept form of the line:

y - 1 = 2x – 6

y = 2x – 5

If you are trying to link the Slope-Intercept form to the Point-Slope form of the line:  The point slope version eliminates one step from using the slope intercept form

In the slope intercept form, you plug in the point and the slope and solve for b—
AND THEN rewrite the equation using the intercept that you found.

In point slope form, once you plug in the point and slope, you just simplify and the equation is already done!

IF you don’t have the slope, you will need to compute it with the formula—just like you did with Slope Intercept FORM

The biggest difference is that any point is plugged into this form, while  the Slope- Intercept Form focuses on the ONE specific point on the y axis 

Algebra (Period 5)

Chapter 4-2 Writing Equations in Slope-Intercept Form

We usually use the slope- intercept form of the line as our  ”template”

We know that y = mx + b so we can substitute that in what we know (what the problem gives us as information) and solve for whatever we are missing

It helps to memorize this little rhyme (Mrs Sobieraj made it up!)

Oh mystery line,
What could you be?
If I could just find you,
 y = mx + b
First I’ll find m,
Then I’ll find b
Then I’ll put it all together
And I will see: 
y = mx + b

The rhyme has 3 steps and usually you will have 3 steps or questions to ask yourself:

1) Do I have the slope (m)? If not find it by using the slope formula or counting it if you have the graph—(carefully pick two sets of integer points)

2) Do I have the y- intercept (b) ? If not, find it by plugging in a point and the slope and solving for b or if you have the graph, just read it on the y axis.

3) Remember: Put it all together in ONE equation at the end!

There are FIVE general cases of mystery lines

First Case: 
You are given the slope and the y intercept (that is the easiest case) 
For example:  you are given m = 3/2 and b = - 7/5

Just plug in to the generic slope intercept equation

y = (3/2)x – 7/5

Second Case:
You have a graph of a line and need to determine the equation
Look at the graph and find 2 easy points to use to find the slope ( make sure they are integers) If the y intercept is not an integer—then follow the FOURTH CASE (below) completely!

Put the information together in y = mx + b form

Third Case:
You are given a point and the slope and need to find the intercept ( b)
Example: ( 3, 1)is a point on the line and m = 2

Plug in the point and the slope and find b

That is, start with y = mx + b 
You have a point ( 3, 1) plug it in to that equation:

1 = (2)(3) + b
1 = 6 + b

-5 = b or
b = -5

Now put it altogether with the given slope of m = 2 and the y intercept ( b) which you just found

y = 2x – 5

Fourth Case:
You are given a point and the y intercept and need to find the slope > Let’s use the point ( 3, 1) again but this time you are given b = 2

Again you can use y = mx + b . This time, however you are solving for m ( the slope)
1 = 3m + 2
-1 = 3m

-1/3 = m
m = -1/3

Again, NOW put it all together with the given intercept and the slope you just found
 y = (-1/3)x + 2

Fifth Case:
You are given 2 points on a line and need to find the slope and the y intercept
Example: ( 1, 3) and ( -2, -3) are 2 points on the line

You first need to find the slope using the formula
m = change in y/ change in x

m = (-3 -3)/(-2-1)  or  (3--3)/(1--2)  which really is (3+3)/(1+2) or
6/3 = 2

Now plug the slope in with one ( you get to pick—it will work with either) of the points and find the intercept, b

3 = 2(1) + b

3 = 2+b

b = 2

Finally put it all together

y = 2x +1

Algebra Honors (Period 4)

Chapter 4-2 Writing Equations in Slope-Intercept Form

We usually use the slope- intercept form of the line as our  ”template”

We know that y = mx + b so we can substitute that in what we know (what the problem gives us as information) and solve for whatever we are missing

It helps to memorize this little rhyme (Mrs Sobieraj made it up!)

Oh mystery line,
What could you be?
If I could just find you,
 y = mx + b
First I’ll find m,
Then I’ll find b
Then I’ll put it all together
And I will see: 
y = mx + b

The rhyme has 3 steps and usually you will have 3 steps or questions to ask yourself:

1) Do I have the slope (m)? If not find it by using the slope formula or counting it if you have the graph—(carefully pick two sets of integer points)

2) Do I have the y- intercept (b) ? If not, find it by plugging in a point and the slope and solving for b or if you have the graph, just read it on the y axis.

3) Remember: Put it all together in ONE equation at the end!

There are FIVE general cases of mystery lines

First Case: 
You are given the slope and the y intercept (that is the easiest case) 
For example:  you are given m = 3/2 and b = - 7/5

Just plug in to the generic slope intercept equation

y = (3/2)x – 7/5

Second Case:
You have a graph of a line and need to determine the equation
Look at the graph and find 2 easy points to use to find the slope ( make sure they are integers) If the y intercept is not an integer—then follow the FOURTH CASE (below) completely!

Put the information together in y = mx + b form

Third Case:
You are given a point and the slope and need to find the intercept ( b)
Example: ( 3, 1)is a point on the line and m = 2

Plug in the point and the slope and find b

That is, start with y = mx + b 
You have a point ( 3, 1) plug it in to that equation:

1 = (2)(3) + b
1 = 6 + b

-5 = b or
b = -5

Now put it altogether with the given slope of m = 2 and the y intercept ( b) which you just found

y = 2x – 5

Fourth Case:
You are given a point and the y intercept and need to find the slope > Let’s use the point ( 3, 1) again but this time you are given b = 2

Again you can use y = mx + b . This time, however you are solving for m ( the slope)
1 = 3m + 2
-1 = 3m

-1/3 = m
m = -1/3

Again, NOW put it all together with the given intercept and the slope you just found
 y = (-1/3)x + 2

Fifth Case:
You are given 2 points on a line and need to find the slope and the y intercept
Example: ( 1, 3) and ( -2, -3) are 2 points on the line

You first need to find the slope using the formula
m = change in y/ change in x

m = (-3 -3)/(-2-1)  or  (3--3)/(1--2)  which really is (3+3)/(1+2) or
6/3 = 2

Now plug the slope in with one ( you get to pick—it will work with either) of the points and find the intercept, b

3 = 2(1) + b

3 = 2+b

b = 2

Finally put it all together

y = 2x +1