Slope

Traditional Slopes:Positive Slope:

This is a slope that increases as you move from left to right on a

coordinate plane…think of riding a ski lift up a

mountain.

Negative Slope:

This is a slope thatdecreases as you move fromleft to right on a coordinate

plane…think of downhillskiing.

Zero Slope:

This is a slope of zero which forms a horizontal

line…think of cross-country skiing.

Undefined Slope:

This is a vertical line with a non-existent

slope…think of extreme skiing.

Slope Formula:

The y2 and y1 represent the 2 y-values in the ordered pairs, and x2 and x1 represent the x-values.

Remember, because slope is “rise over run” the y-values need to be in the numerator and the x-

values go in the denominator!

2 1

2 1

y ym

x x

Example #1: Plotting points and finding slopes

Find the slope of the segment with endpoints at (2, 3) and (8, 6) by plotting the points and counting the rise over the run.

Up 3

Over 6

3 1 or

6 2m m

Example #1: Plotting points and finding slopes

Using these same points, let’s plug them into the slope formula to see what we get:

(2, 3) and (8, 6)

6 3

8 2m

3

6m

1

2m

Lots More Practice!Using the slope formula, find the slopes for the following

sets of points:

1. (6, 12) and (5, 4)

2. (3, -1) and (-3, 2)

3. (7, -2) and (3, -2)

4. (7, 12) and (7, 8)

8m

4

0m which givesm undefined

1

2m

00

4m which givesm

So What Do These Slopes Look Like Side-by-Side?

We are going to take a short trip to the National Library of Virtual Manipulatives to check out some graphs and see

their differences in slopes!http://nlvm.usu.edu/en/nav/frames_asid_109_g_4_t_2.html?open=activities&from=category_g_4_t_2.html

Once you arrive at the page click on the “functions” tab and enter in different equations for f(x), g(x), and h(x) and hit “graph” after each one. All three graphs will

appear so we can notice some differences with different slopes!

Midpoint

• A midpoint is the point on a line segment that is the same distance from both endpoints.

– (It’s the exact middle of the line segment)

A CB

What is the Midpoint?

Coordinate Midpoint

• You can find the coordinates of the

midpoint using this formula:

Midpoint =

How do you find the midpoint of a segment?

x x y y1 2 1 2

2 2

FHG

IKJ,

Find the coordinates of the midpoint

(-2,-5)

(6,13)

Find the Midpoint

• Ex. 1) Find the midpoint of the segment with the endpoints (-2,-5) & (6,13).

x1 y1 x2 y2

The midpoint between (-2, -5) and (6, 13) is (2, 4)

2 6 5 13 4 8, ,

2 2 2 2

2,4

Find the Midpoint

(-2,-5)

(6,13)

(2, 4)

Ex. 2)Find the midpoint of the segment with the endpoints (14,-7) & (3,18).

14 3

2

7 18

2

17

2

11

2

FHG

IKJFHG

IKJ, ,

The midpoint between (14, -7) and (3, 18) is

(17/2, 11/2)

x x y y1 2 1 2

2 2

FHG

IKJ,

Find the midpoint of (5,4) and (3,6)

x x y y1 2 1 2

2 2

FHG

IKJ,

5 3 4 6,

2 2

8 10,

2 2

Ex #3

4,5

What if we have the midpoint and one endpoint and want to find the other

endpoint.1 2

2 x

x xM

Mx = x value of the midpointAnd

My = y value of the midpoint

1 2

2 y

y yM

Ex 1 Midpoint (0, -6) Endpoint (7, -12)Find the other endpoint

(0, -6)

(7, -12)

(?, ?)

Midpoint (0, -6) Endpoint (7, -12)

For the x value:

7

20

x 27

20 2

x

7 0 x -7 -7

X = -7

Step 1: plug values

Step 2: Multiply both sides by 2

Step 3: Solve for x Step 4: Solution

Now do the same for the y value.

12

26

y

Y = 0

So the other endpoint is: (-7, 0)

12 12y

Midpoint (0, -6) Endpoint (7, -12)So the other endpoint is (-7, 0)

(0, -6)

(7, -12)

(-7, 0)

Ex 2) Find the other endpoint

Midpoint: (7,4) Endpoint: (2,4)

2

27

x

4

24

y

so

so

so

so

X = 12

Y = 4

The other endpoint is (12, 4)

1 2

2 y

y yM

1 2

2 x

x xM

Ex 3) Find the other endpointMidpoint (-9, 7) Endpoint (15, -7)

Answer: (-33, 21)

1 2

2 y

y yM

15

29

x

7

27

y

X = -33

Y = 21

1 2

2 x

x xM

Summary:

Write 3 things about slope that you learned today from this lesson.