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8-14-15 & 8-17-15AM1.1b To Find Slopes and Equations of Lines

T. J.

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Active Learning Assignment?

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Puff, Puff, positive slope

Nice, … negative slope

zzzzzzz, … zero slope

Oooooh nooooo, no slope

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I. Lesson: Slope of a line: a n umber measuring the steepness of a line relative to the x-axis. The slope of the line is written as rise over run. This ratio is a rate of change. That means it is the change in y over the change in x. The variable is “m”.

risem

run The “rise” can be up or down (+ or –),

but it always runs to the right.

Why do we use “m” for slope?

Positive slope

Negative slope

5

ym

x

1 2 2 1

1 2 2 1

y y y ym or

x x x x

1If the slope is , that means :

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y rise x run

I prefer:*

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Example: Find the slope between (2, -3) and (5,-6).

First, ALWAYS write:y

mx

Then, fill it in,

putting negative numbers in parentheses.

31

3

(-3) (-6)

2 5

What would happen if you used y2 first?

ym

x

31

3

(-6) (-3)

5 2

Tah dah!!!!!!!!

!!!!!Notice, ORDER is important!

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Try: Find the slope between (-1, 7) and (-2,11).

ym

x

44

1

7

(-1) (-2)

11

= 0 (Zero slope, OK)

What if you end up with:0

?7

= No slope (Undefined)3

?0

Very Important

!

or

ym

x

44

1

11

(-2) (-1)

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Idea from Alyssa Bonacci, Class of 2016

N

K

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y = mx + b

-2 = 4(5) + b-2 = 20 + b

-22 = b y = 4x – 22

II. Lesson: Find the equation of the line that:

Passes through (5,-2) and has a slope of 4.

The answer:

Try: Find the equation of the line that: Passes through (-2,-6) and has a slope of

1

2y = mx + b

-6 = (-2) + b

-6 = -1 + b

-5 = b

1

2 1y x 5

2

The answer:

(x, y) m

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III. Lesson: Find the equation of the line that:Passes through (4,-1) and is parallel to y = 2x – 12

What information did we have last time? How can this problem compare to it?

If two lines are parallel, they have the same slope! Soooo…

y = mx + b

-1 = 2(4) + b-1 = 8 + b-9 = b

y = 2x – 9

m = 2

The answer:

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When the equation is in the format y = mx +b. The “m” represents the slope. So, if we have

y = 7x + 52 then the slope is 7!

What is the slope for 2x – 3y = 41?

–2x –2x

–3y = –2x + 41

3y 2x 41

3 3

2 41

y x3 3

2Thus m

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There’s a shortcut. What could it be?

Ax + By = C

Am

B

*

When you have Ax + By = C

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Try: Find the equation of the line that:Passes through (2, 6) and is parallel to 3x + 2y = 25

First, find the slope:A 3 3

mB 2 2

y = mx + b

6 = (2) + b

6 = -3 + b

9 = b

3

2

3

y x 92

The answer:

12

What can we say about two lines that are perpendicular?

1. One slope is positive and one is negative

2. The rises and runs are reciprocals.

3. Thus the perpendicular slopes are opposite reciprocals.

1

m

+

–

12

1-2

1 2vs

2 1

Symbol:

1

2

2

1

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IV. Lesson: Find the equation of the line that:Passes through (2,5) and is perpendicular to

2y x 18

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1. What is the given slope?

2. What slope do we want?

1 2

m 7

7m

2

y = mx + b

5 = (2) + b

5 = -7 + b

12 = b

7

2

7y x 12

2

Notice the two slopes

The answer:

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Try: Find the equation of the line that: Passes through (9, -4) and is perpendicular to

y = mx + b

-4 = 9 + b

-4 = 15 + b

-19 = b

y = x – 19

3y x 11

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1. What is the given slope?

2. What slope do we want?

1 3

m 5

5m

3

5

3 5

3

The answer:

15

ym

x

84

2

7 (-1)

5 3

V. Lesson: Find the equation of the line that: Passes through (5,7) and (3, -1)

First, find the slope:

Now, we have a slope, which point do we use?

Since they are BOTH on the line, it does not matter!!!

7 = 4(5) + b7 = 20 + b

-13 = b

y = 4x – 13

-1 = 4(3) + b-1 = 12 + b

-13 = b

y = mx + b

The answer:

or

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Try: Find the equation of the line that:Passes through (-6, -5) and (-4, 3)

ym

x

84

2

(-5) 3

(-6) (-4)

y = mx + b

-5 = 4(-6) + b-5 = -24 + b19 = b y = 4x + 19

The answer:

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Active Learning Assignment:

I. P. 11 Written Exercises, #1-6

II. “Find the Equation that:” handout. (The handout and answers to this section will be posted on Weebly.)

III. WOW: • Floss your teeth every day• Forgive others (and yourself)• Cherish your Friends