**S**

P

E

O

L

**5.2**

BELL RINGER

1.) The band is trying to raise money to take a field trip to the Rock and Roll Hall of Fame. They decide to sell sweatshirts. The equation for the amount of money a that they will make for selling t sweatshirts is a = 22t - 350. In order to make at least $2100, how many sweatshirts do the band members need to sell?

A.) 112

B.) 111

C.) 80

D.) 79

VOCABULARY

a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable.

rate of change

slope

rise

run

slope formula

the ratio of rise to run for any two points on a line

the difference in the y-values of two points on a line

the difference in the x-values of two points on a line

the ratio of the difference in y-values to the difference in x-values between any two different points on a line; this formula can be represented as the following:

m = y - y

x - x

Where m is the slope and (x ,y ) and (x , y ) are the points.

1

1

2

2

2

2

1

1

**x**

**y**

Which of these segments has the largest slope? The smallest?

This part of the graph has a "positive" slope because it is going up, or increasing, from left to right.

The segment that is changing the fastest has a larger slope.

We will learn how to measure slope and apply a value to it a little later today.

This part of the graph is increasing the fastest, and so it has the largest slope.

This segment is increasing

so it is positive, but it is not increasing as fast as another part of the graph

This part of the graph has a "negative" slope because it is decreasing, or moving down as we look from left to right.

The segment that is decreasing the fastest has the least slope, since its slope is more negative than other parts of the line.

This segment is decreasing so it is negative, but it is not decreasing as fast as another part of the graph

This part of the graph is decreasing the fastest the "most negative" slope.

This segment appears to be horizontal.

If it is horizontal, or "flat" then would it be

positive (like the increasing parts)

OR

negative (like the decreasing?)

What kind of slope does a "flat" line have?

A horizontal, or "flat," line doesn't have a positive or negative slope.

It has a slope in between positive and negative: a slope of 0.

horizontal line = slope of 0.

(-2, -2)

(3, 4)

Let's say for a moment a line passes through the points (-2, -2) and (3, 4).

Would the slope of that line be positive or negative?

What would that slope be?

We have to ways to find the slope of

this line...

One way is to plug the two points into the slope formula:

m = = =

y - y 4 - (-2) 6

x - x 3 - (-2) 5

The line has a slope of .

6

5

(-2, -2)

(3, 4)

point 1

point 2

x

y

x

y

1

1

2

2

+6 rise

+ 5 run

The second way to find this slope is to

measure the rise (up or down change)

and the run (right or left change)

on a graph.

Since we read left to right, we will

also measure left to right, then

measure up or down, and

up or down will determine

whether slope is positive

or negative.

So,

either way...

the slope is

6

5 .

So what kind of slope would a

v

e

r

t

i

c

a

l

line have?

Remember that slope is:

rise (change in y)

OVER

run (change in x)

**?**

RISE

RUN

What

is

the

change

in

y?

(-3, 6)

(-3, 0)

What is the change in x?

RATE OF CHANGE AND SLOPE

A rate of change is can be used to find how fast something increases or decreases.

The price of gasoline, for example, changes almost daily, and has changed dramatically over the past 3 decades.

These ups and downs can be represented graphically (that means in graphs...remember people like pictures a lot more than words so graphs are a good thing...)

When rate of change is represented in a graph, it can be measured by the "slope" of the graph.

The change in y is 6.

The change in x is 0.

Since slope is rise over

run, this means the slope is 6 over 0.

Because we cannot divide by 0, the slope of a vertical line is:

**UNDEFINED**

m = =

6 - 0 6

-3 - (-3) 0

**PRACTICE**

Find the slope of a line that passes through the points

(-2, -1) and (4, 2)

Find the slope of a line that passes through the points

(-2, 5) and (4, 5)

Find the slope of a line that passes through the points

(-1, 4) and (-1, -3)

Find the slope of a line that passes through the points

(-6, 3) and (1, -4)

**SOLUTIONS**

m = 0

slope is undefined.

m = -1

m = 1/2

1

1

2

2