Slope

What is slope?Slope is the steepness of a line as it moves from LEFT to RIGHT.

Slope is the ratio of the rise, the vertical change, to the run, the horizontal change of a line.

The slope of a line is always constant (it never changes) no matter what 2 points on the line you choose.

PictoriallySLOPES ARE READ FROM LEFT TO RIGHT

Are these pictures of positive slopes or negative slopes?(the lines fall from left to right)

Is this a picture of a positive or a negative slope?(the line rises from left to right)

Slope = =

Slope is denoted as the letter ‘m’ and it measures the steepness of a line.

Slope = What does it mean for a line to have a slope of 5/7? It means that for every 5

units of change in the y-direction, there is a change of 7 units in the x-direction.

To find the slope of a line create the right triangle between the two points.

Graphically finding slope

Slope = = =

Slope, m =

This means that for every 5 units of change in the y-direction, there is a change of 4 units in

the x-direction.

Vertical & Horizontal Lines

A Vertical line means an undefined slope – there is NO change in x. (Your slope will have a denominator of zero.)Example: picking two points

m = = undefined

A Horizontal line means a slope of zero – there is NO change in y. (Your slope will have a numorator of zero.)Example: picking two points

m = = zero

How do these pictures help us?

Real world slopes

Find the slope or pitch of the roof.

Find the slope of theBoat ramp.

3 ways to determine slope

1. Graphically

2. Using a table

3. Using two points

Slope-intercept form of an equation

The slope-intercept form of an equation is y = mx + b, where m is the slope and b is the y-

intercept.The y-intercept of a lineis the y-coordinate of thepoint where the line crosses the y-axis. It occurswhen x = 0. It’s coordinate will be (0,y)

y-intercept is at (0,2)

Finding the slope (m) and y-intercept (b) from a graph and write the equation in slope-intercept form

Step 1: Draw a right triangle. Step 2: Calculate the Step 3: Find the y-intercept (b), where the line cross the y-axis?Step 4: Write the slope-intercept equation.y = mx + b by plugging in the values of m andb.

Find slope (m) and y-intercept from an equation

Given an equation in slope-intercept form:y = 2x + 1 slope (m) = ?

y-intercept (b) = ? y = -x + 1 slope (m) = ?

y-intercept (b) = y = 4x – 5 slope (m) =

y-intercept (b) = ?

Given the equation of a line in slope-intercept form, graph the line.

y = 2x + 1y = mx + bm = b = 1

y-intercept is at (0,1)slope is = up 2 and to theright 1

Given the equation of a line in slope-intercept form, graph the line.

1. y = 3 x + 22. y = x - 13. y = - x + 3

Writing an equation of a line in standard form

Ax + By = C (where A is a positive integer and B and C are integers)

Why do we care about standard form?

Recall that the slope-intercept form of a line is: y = mx + b. To change this into standard form, we start by moving the x-term to the left side of the equation. This is done by subtracting mx from both sides. We now have the equation, -mx + y = b. The coefficient of the x-term should be a positive integer value, so

we multiply the entire equation by an integer value that will make the coefficient positive. This gives us the standard form: Ax + By = C

Going from slope-intercept to standard form

Write these equations in standard form y = -3x + 6

First, we need to move the x-term to the left side of the equation so we add 3x to both sides. Doing this gives us: 3x + y = 6. Here, the coefficient of the x-term is a positive integers and all other values are integers, so we are done.

y = 2x + 7Again, start by moving the x-term to the left. Subtract 2x from both sides to get: -2x + y = 7. We need the x-term to be positive, so multiply the equation

by -1 to get our answer.

Write the equation of the line with a slope of (-3/4 ) that passes through the point (0,6) in standard form

First, using the given information, m = (-3/4) and b = 6, we use slope-intercept form, y = mx + b to start. Substitution of m and b

gives us: y = (-3/4)x + 6. To convert to standard form. First add (3/4)x to both sides to get:

(3/4)x + y = 6. Finally, we must get rid of the fraction by multiplying by multiplying the whole equation by 4. This multiplication yields the answer which is: 3x + 4y = 24.

When you do not have a graphical representation

Using a table OR

Use:

With the table: With two points:

Using two points

Slope = =