Slope describes the slant and direction of a line.

What is Slope? The slope of a line refers to the steepness of the line. The slope also determines whether the linear function will increase or decrease from left to right.

Geometric Interpretation of Slope Slope measures the steepness of a line and is given by the formula: x y O run rise rise run slope =

Given two points (x1, y1) and (x2, y2), the formula for the slope of the straight line going through these two points is:

The Notation The subscripts indicate that you have a "first" point and a "second" point .  Slope is sometimes referred to as "rise over run“. The fraction consists of the "rise" (change in y, going up or down) divided by the "run" (change in x, going to the left or right).

Alien Explanation!

How do you find the slope of a line? The slope is looked at as a fraction that represents m = the change in the y direction the change in the x direction The variable used to represent slope is m. If the slope ends up as a fraction, LEAVE it as a fraction in simplest form. Do not change it to a decimal!

The two points shown are: (0, -4) and (-3, -6) The two points shown are: (0, -4) and (-3, -6). Now we have two points we can put them in the slope formula:

Example #2

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

Horizontal Lines The points (-3, 4) and (5, 4), the slope is: For every horizontal line:  a slope of zero means the line is horizontal, and a horizontal line means you'll get a slope of zero. The equation of all horizontals lines are of the form: y = “a number“ (ex. y = 4)

Vertical Lines Now consider the vertical line of the equation x = 4: A vertical line will have no slope, and “the slope is undefined" means that the line is vertical.  The equation of all vertical lines are of the form x = “a number“ (ex. x = 4). 

There are 4 types of Slope 1. Positive Slope A line that rises from left to right 2. Negative Slope A line falls from left to right 3. Zero Slope A line is horizontal 4. Undefined Slope The line is vertical

Find the slope of a line that contains points: (5, 4) and (5, 2). This slope is undefined.

First pick any two points on the line. (5,6) (-4,-2) First pick any two points on the line. Then find the coordinates of the points and use them in the slope formula.

Find the Slope Red (3, 9) Blue (11, 2) Green (5, -2)

Q & A If I am given a line, how do I find the slope? Pick any two points on the line and use the slope formula.

Q & A If I pick two different points than someone else won’t I get a different answer for the slope? No, the math will be different, but the answer will be the same.

Things to Remember If the slope ends up positive, what will the line look like? If the slope ends up negative, what will the line look like?

Things to Remember The slope formula

Key Skill Vanessa starts to ski at the top of a 400 foot mountain. After 15 minutes she is at 250 feet. Graph her location on the mountain, then find the slope of the line. What are the 2 sets of (x,y) coordinates? (0, 400) (15, 250)

Key Skills y Graph the points (0, 400), (15, 250) 1000 Graph the points (0, 400), (15, 250) 500 Does the direction of the line show a positive or negative trend? x -20 -10 10 20 -500 Negative. -1000 Find the slope.

Rules and Properties Slope as a Difference Ratio Given two points with coordinates (x1, y1) and (x2, y2), slope is given by the formula: y2 – y1 x2 – x1 m = x y O (x1, y1) (x2, y2)

Key Skill . m = y2 – y1 (0, 400) (15, 250) x1 y1 x2 y2 x2 – x1 (0, 400) (15, 250) x1 y1 x2 y2 x2 – x1 Label the coordinates m = 250 – 400 -150 = = -10 15 - 0 15 Substitute into the formula The slope = -10

y Key Skills 1000 Notice that the slope is -10 and the line is in a negative direction. 500 x -20 -10 10 20 -500 -1000

Key Skills TRY THIS Edgar deposited $100 in the bank. After 6 weeks he deposited a total of $220. Graph the amount of money he has in his account then find the slope. What are the 2 sets of (x, y) coordinates? (0, 100) (6, 220)

Does the direction of the line show a positive or negative trend? x TRY THIS Key Skills y 250 Graph the points (0, 100), (6, 220) 125 Does the direction of the line show a positive or negative trend? x -10 -5 5 10 -125 Positive. -250 Find the slope.

Key Skill . m = y2 – y1 (0, 100) ( 6, 220) x1 y1 x2 y2 x2 – x1 (0, 100) ( 6, 220) x1 y1 x2 y2 x2 – x1 Label the coordinates m = 220 – 100 120 = = 20 6 - 0 6 Substitute into the formula The slope = 20

Notice that the slope is 20 and the line is in a positive direction. TRY THIS Key Skills y 250 Notice that the slope is 20 and the line is in a positive direction. 125 x -10 -5 5 10 -125 -250

Graph the 2 points and find the slope. TRY THIS Key Skills y 10 Graph the 2 points and find the slope. 5 (-5, 5) (5, 5) x -10 -5 5 10 -5 -10

Key Skill . m = y2 – y1 (-5, 5) ( 5, 5) x1 y1 x2 y2 x2 – x1 (-5, 5) ( 5, 5) x1 y1 x2 y2 x2 – x1 Label the coordinates m = 5 – 5 = = -5 - 5 -10 Substitute into the formula The slope = 0

The slope of all horizontal lines are 0. TRY THIS Key Skills y 10 The slope of all horizontal lines are 0. 5 x -10 -5 5 10 -5 -10

Graph the 2 points and find the slope. TRY THIS Key Skills y 10 Graph the 2 points and find the slope. 5 (3, 6) (3, -6) x -10 -5 5 10 -5 -10

Key Skill . m = y2 – y1 (3, 6) ( 3, -6) x1 y1 x2 y2 x2 – x1 (3, 6) ( 3, -6) x1 y1 x2 y2 x2 – x1 Label the coordinates m = -6 – 6 -12 = = undefined 3 - 3 Substitute into the formula The slope = undefined

The slope of all vertical lines are undefined. TRY THIS Key Skills y 10 The slope of all vertical lines are undefined. 5 x -10 -5 5 10 -5 -10

The slope of a line can be defined as: ∆y This means the change in y divided by the change in x. ∆x M = 2 This means that as y rises by 2, x runs 3 to the right. 3

A line has a slope of -3. It contains a point (1, 2). Graph the line. Key Skills y A line has a slope of -3. It contains a point (1, 2). Graph the line. 10 5 m = -3 x 1 -10 -5 5 10 To find the new point, add -3 to the y, add 1 to the x. -5 (2, -1), now graph. -10

To find the new point, add 1 to the y, add 2 to the x. TRY THIS Key Skills y A line has a slope of 1/2. It contains a point (3, -2). Graph the line. 10 5 m = 1 x 2 -10 -5 5 10 To find the new point, add 1 to the y, add 2 to the x. -5 (5, -1), now graph. -10

Rules and Properties Geometric Interpretation of Slope Slope measures the steepness of a line and is given by the formula: rise run slope = x y O run rise

Key Skills Find the slope of a line by using the rise and run. slope = m = = 3 15 5

Key Skills TRY THIS Find the slope of a line by using the rise and run. rise: -4 run: -16 slope = m = = -4 -16 1 4