PRE-ALGEBRA

PRE-ALGEBRA

Lesson 8-3 Warm-Up

PRE-ALGEBRA

What is “rate of change”

How can you find the “rate of change”?

rate of change: a unit rate that tells how fast the dependent variable (y) is changes in relation to the dependent variable (y) – In other words, “x changes _?_ much for every change of 1 in y”

Since the independent variable is plotted on the x-axis (the x’s) and the dependent variable is plotted on the y-axis (the y’s), we can find the rate of change by using the following rule:

Rule:

Since rate of change is a linear function (forms a line when graphed), you can use two point on the line to find the vertical and horizontal changes, such as (x1, y1) and (x2,, y2). To find the vertical change, find the difference in the y terms (y2,- y1), and to find the horizontal change, find the difference in the x terms (x2,- x1)

Example: The following graph shows the altitude of an airplane as it’s coming in for a landing. Find the rate of change.

Slope and y-Intercept (8-3)

PRE-ALGEBRA

The rate of change is , which means the airplane descends

feet every second.

Slope and y-Intercept (8-3)

PRE-ALGEBRA

What is “slope”

How can you find the slope?

slope: the rate of change of a line on a graph in a unit rate form (how much the y changes for every 1 change in x)

Rule:

You only need two points on the line to find slope (just like you do for rate of change). You can either: 1. divide the difference of the y values by the difference of the x values, or 2. simply divide the rise (vertical distance between the two points) and the run (horizontal distance between the two points) to find the slope as in the following examples.

Note: When using the subtraction method, make sure you start with the same point for the y’s on top and the x’s on the bottom.

Example: Find the slope of the line.

Slope and y-Intercept (8-3)

PRE-ALGEBRA

slope = riserun

Example: Find the slope of the following line.

Method 1: Divide the difference of the y’s

by the difference of the x’s.

Step 1: Find two pints on the line.

(-1, 2) and (1, -2) are on the line.

Step 2: slope =

= = = -2 Start with the same point for x and y. In this

case, we started with (1, -2).

Method 2: Divide the rise and run.

Slope = Find the rise and run from one point to another.

= -2 The rise from the two point selected is -4 and the run is 2.

The slope of the line is -2.

LESSON 8-3

Additional Examples

Slope and y-intercept

change in y’s change in x’s

-2 – (2) 1 – (-1)

-42

-42

riserun

PRE-ALGEBRA

How do you determine whether a slope (the rate of change) is positive, negative, zero, or undefined (impossible)? can you tell if a slope is positive, negative, 0 (no slope), or undefined (impossible)?

Slope and y-Intercept (8-3)

PRE-ALGEBRA

Find the slope of each line.

a. b.

slope = = = 4riserun

41

slope = = = –2riserun

–63

Slope and y-interceptLESSON 8-3

Additional Examples

PRE-ALGEBRA

Find the slope of the line through each pair of points.

slope = = = =difference in y-coordinatesdifference in x-coordinates

0 – 5–2 – 7

–5–9

59

slope = = = = 0difference in y-coordinatesdifference in x-coordinates

– 3 – (– 3) 4 – (– 2)

06

slope = = = = undefineddifference in y-coordinatesdifference in x-coordinates

3 – (– 1)–2 – (– 2)

40

b. A (–2, –3), B (4, –3)

a. E (7, 5), F (–2, 0)

c. R (–2, –1), S (–2, 3)

Slope and y-interceptLESSON 8-3

Additional Examples

PRE-ALGEBRA

A store sells almonds in bulk for $6 per pound.

Graph the relation (pounds of almonds, cost). Find the

slope of the line through the points on your graph.

Compare the slope to the unit rate .cost

one pound of almonds

Slope and y-interceptLESSON 8-3

Additional Examples

PRE-ALGEBRA

The slope equals the unit rate.

(continued)

Slope and y-interceptLESSON 8-3

Additional Examples

Use two points to find the slope.

The unit rate is

=

$61 pound of almonds

61

Slope = 12 – 62 –1 =

61

6

1

6

1

6

1

PRE-ALGEBRA

What is the “y-intercept”?”?

What is “slope-intercept form”

y-intercept: the point at which a line intercepts, or intersects, with the y-axis.

Example: (0,3) is the y-intercept of the following graph.

Slope-intercept form: a linear equation(forms a line) which include the slope of the line and its y-intercept (where the line crosses the y-axis) – It is written as:

where m (the coefficient of the x term) is the slope and b is the y-intercept (where the line crosses the y-axis)

Slope and y-Intercept (8-3)

PRE-ALGEBRA

How do you graph an equation using slope-intercept form?

To graph an equation using slope-intercept form: 1. plot the y-intercept, 2. use the slow to draw at least one stair step from the y-intercept, 3. plot a point at the end of each stair step, and 4. connect the points with a line. at which a line intercepts, or intersects, with the y-axis.

Example: Graph y = - x + 4.

Step 1: The y-intercept is 4, so start at (0,4)

Step 2: The slope is - which means a rise of -3 (down 2 – the negative always goes with the rise) and a run of 2 (right 2). Create a few stair-steps that go down 3 and right 2.

Step 3: Connect the dots with a line.

Slope and y-Intercept (8-3)

32

32

PRE-ALGEBRA

A ramp slopes from a warehouse door down to a

street. The function y = – x + 4 models the ramp, where x is the

horizontal distance in feet from the bottom of the door and y is the

height in feet above the street. Graph the equation.

15

Step 1 Since the y-intercept is 4, graph (0, 4).

Step 3 Draw a line through the points.

Then move 5 units right to graph a second point.

Step 2 Since the slope is – , move

1 unit down from (0, 4).

15

Slope and y-interceptLESSON 8-3

Additional Examples

PRE-ALGEBRA

Find the slope of the line through each pair of points.

1. A(2, 4), B(–2, –4) 2. F(–5, 1), G(0, –9)

3. Identify the slope and y-intercept of y = – x + 3. Then graph the line.

2 –2

43

– ; 343

Slope and y-interceptLESSON 8-3

Lesson Quiz