Slope and Rate of Change

Rates of change are seen everywhere.

The steepness of the roof of a house is referred to as the pitch of the roof by home builders.

Give one reason why some homes have roofs which have a greater pitch.

There is less snow buildup in the wintertime.

Engineers refer to the rate of change of a road as the grade.

What is the meaning of this sign?

1. Icy Road Ahead2. Steep Road Ahead3. Curvy Road Ahead4. Trucks Entering

Highway Ahead

What does the 7% mean?7% is the slope of the road.

It means the road drops 7 feet vertically for every 100 feet

horizontally.

7%

So, what is slope???Slope is the steepness of a line.

It is the rate it changes going up and down

7 feet

100 feet

SLOPESlope is the ratio of the vertical rise to the horizontal run between any two points on a line. Usually referred to as the rise over run. Slope triangle between two

points. Notice that the slope triangle can be drawn two different ways.

Rise is -10 because we went down

Run is -6 because we went to the left

3

5

6

10

iscasethisinslopeThe

Rise is 10 because we went up

Run is 6 because we went to the right

3

5

6

10iscasethisinslopeThe

Another way to find slope

Write this down—Slope is a measure of how

much a line changes going up and down ( or rise)

for how much it changes going left to right ( or run )

y

x

And thisLinear functions have a constant rate of change

Nonlinear functions do not

Types of Slope

PositiveNegative

Zero

Undefinedor

No SlopeComplete Practice A

If given 2 points on a line, you may findthe slope using the

formula 2 1

2 1

y ym

x x

Write it on your paper

Find the slope of the line through the

points (3,7) and (5, 19).x1 y1

x2 y2

m = 19 – 7

5 – 3 m = 12 2

m = 6

Complete Practice B

Find the missing coordinate if the slope is 6

points (3, ) and (5, 19)

2 1

2 1

y ym

x x

19 y5 3

61

Solve by cross multiplying

19 y 12

2

19 19y 7 y 7

7y

Now try Practice C

Linear equations -only have 1 or 2 variables -the variables are not multiplied together, - variables have a power of 1

Which of the following are linear?

2x 3y 12

4y 7x 2 y 3x 1 y 2 3(x 4)

y 7x 17

yes

yesyes

yesyesyes

Yes, they all are

Write this down. 3 common forms for linear equations

Slope intercept form y = mx + b

Point slope form

Standard form Ax + By = C Today we are going start by talking about slope intercept form

1 1y y m(x x )

SLOPE-INTERCEPT FORM OF A LINEThe slope intercept form of a line is y = mx + b, where “m” represents the slope of the line and “b” represents the y-intercept.

When an equation is in slope-intercept form the “y” is always on one side by itself. It can not be more than one y either.

We have talked about what slope is….

What is a y intercept?

Y AND X INTERCEPTS

The x-intercept is the x-coordinate of a point where the graph crosses the x-axis.

The y-intercept is the y-coordinate of a point where the graph crosses the y-axis.

The x-intercept would be 4 and is located at the point (4, 0).

The y-intercept is 3 and is located at the point (0, 3).

You can write the slope intercept form of a line….

If you know the slope and the y intercept. Just substitute them in

Find the slope intercept equation for graphs with the following:

Slope y intercept

4 10 ½ -9 ¾ 0 0 3

y 4x 10

y mx b

12y x 9 34y x

y 10Go ahead and do practice D

Determine the slope of the line.

The line is decreasing (slope is negative).

2

-1

rise

run2

1 2

Find points on the graph. Use two of them and apply rise over run.

What is the slope of a horizontal line?

The line doesn’t rise!

m 0

number0

All horizontal lines have a slope of 0.

y 3

What is the slope of a vertical line?

The line doesn’t run!

All vertical lines have an undefined slope.

m number

0undefined

x 2

Remember the word “VUXHOY”

Vertical linesUndefined slopeX = number; This is the equation of the line.

Horizontal linesO - zero is the slopeY = number; This is the equation of the line.

Complete practice E

• Identify whether the slope is + or –• Find 2 points on the line• Find the slope by using the rise over run• Give the slope the correct sign• Find the y intercept, what points does the line

cross the y axis?• Put the slope and y intercept together to write

a slope intercept equation

Slope-intercept graphingSteps1.Make sure the equation is in slope-intercept form.2.Identify the slope and y-intercept.3.Plot the y-intercept.4.From the y-intercept use the slope to get another point to draw the line.

1. y = 3x + 22. Slope = 3 (note that this means the

fraction or rise over run could be (3/1) or (-3/-1). The y-intercept is 2.

3. Plot (0, 2)4. From the y-intercept, we are going

rise 3 and run 1 since the slope was 3/1.

Complete practice F

• Plot the y intercept• Use rise over run to plot 2 more points• Connect the points• Put an arrow at each end

Direct VariationsAre linear equations that have a

y intercept of 0These are given a special form y = kxThe k is the same slope, but called the

constant of variationThese graphs are easily identified because

they always pass through the origin Complete practice G