Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1 1 Section 3.3 Slope Copyright © 2013, 2009,...

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

Section 3.3

Slope

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1


Objective #1 Compute a line’s slope.


Slope

The change in the y values is the vertical change or the “Rise”.

The change in the x values is the horizontal change or the “Run.”

m is commonly used to denote the slope of a line. The slope of a line is related to its steepness.

When the slope is positive, the line rises from left to right.

When the slope is negative, the line falls from left to right.


x1 x2

y1

y2

Run

x2 – x1

Rise

y2-y1

Visual Representation of Slope


Slope of a Line

The slope of the line through the two distinct points and is m, where

12

12

in Change

in Change

xx

yy

x

ym

),( 11 yx ),( 22 yx

Note that you may call either point However, don’t subtract in one order in the numerator and use another order for subtraction in the denominator - or the sign of your slope will be wrong.

1 1( , ).x y


Slopes of Lines

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Find the slope of the line passing through the pair of points. Then explain what the slope means.

.2,4, and 8,7,Let 2211 yxyx

11

6

74

82

74

82

in Change

in Change

12

12

xx

yy

x

ym

Remember, the slope is obtained as follows:

This means that for every six units that the line (that passes through both points) travels up, it also travels 11 units to the right.


Slopes of Lines

Possibilities for a Line’s SlopePositive Slope Negative Slope

Zero Slope Undefined Slope


Objective #1: Example

1a. Find the slope of the line passing through the pair of points: 4, 2 and 1,5 .

Let 1 1 2 2, 4, 2 and , 1,5 .x y x y

2 1

2 1

Change in 5 ( 2) 7 7Change in 1 4 5 5

y yym

x x x

The slope is 75

.

Since the slope is negative, the line falls from left to right.



1a. Find the slope of the line passing through the pair of points: 4, 2 and 1,5 .

Let 1 1 2 2, 4, 2 and , 1,5 .x y x y

2 1

2 1

Change in 5 ( 2) 7 7Change in 1 4 5 5

y yym

x x x

The slope is 75

.

Since the slope is negative, the line falls from left to right.



1b. Find the slope of the line passing through 6,5 and 2,5 or state that the slope is undefined. Indicate if the line is horizontal or vertical.

Let 1 1 2 2, 6,5 and , 2,5 .x y x y

2 1

2 1

5 52 604

0

y y

x x

Since the slope is 0, the line is horizontal.

Change in Change in

ym

x



1c. Find the slope of the line passing through 1,6 and 1,4 or state that the slope is undefined. Indicate if the line is horizontal or vertical.

Let 1 1 2 2, 1,6 and , 1,4 .x y x y

2 1

2 1

4 61 12

0

y y

x x

Because division by 0 is undefined, the slope is undefined. Since the slope is undefined, the line is vertical.

Change in Change in

ym

x


Objective #2 Use slope to show that lines are parallel.


Parallel Lines

Slope and Parallel Lines1) If two non-vertical lines are parallel, then they have the same slope.

2) If two distinct non-vertical lines have the same slope, then they are parallel.

3) Two distinct vertical lines, each with undefined slope, are parallel.


Parallel Lines



2. Show that the line passing through (4, 2) and (6, 6) is parallel to the line passing through (0, 2) and (1, 0).

Slope of line through (4,2) and (6,6):

Change in 6 2 4

2Change in 6 4 2

ym

x

Slope of line through (0,–2) and (1,0):

Change in 0 ( 2) 2

2Change in 1 0 1

ym

x

Since their slopes are equal, the lines are parallel.


Objective #3 Use slope to show that lines are perpendicular.


Perpendicular Lines



3. Show that the line passing through ( 1,4) and (3,2) is perpendicular to the line passing through ( 2, 1) and (2,7).

Line through (–1,4) and (3,2):

Change in Change in

ym

x 2 4

3 ( 1)

2412



Line through (–2,–1) and (2,7):

Change in Change in

ym

x

7 ( 1)2 ( 2)

842

Since the product of their slopes is 12 1,

2

the lines are perpendicular.


Objective #4 Calculate rate of change in applied situations.


Slope as Rate of Change

Slope is defined as the ratio of a change in y to a corresponding change in x. It tells how fast y is changing with respect to x. Thus, the slope of a line represents its rate of change.



4. Use the ordered pairs in the figure shown to find the slope of the line segment that represents men. Express the slope correct to two decimal places and describe what it represents.



Let 1 1 2 2, 1990,9.0 and , 2008,14.7 .x y x y

2 1

2 1

Change in 14.7 9.0 5.70.32

Change in 2008 1990 18

y yym

x x x

The number of men living alone increased at a rate of 0.32 million per year. The rate of change is 0.32 million men per year.

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1 1 Section 3.3 Slope Copyright © 2013, 2009,...

Documents

Transcript of Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1 1 Section 3.3 Slope Copyright © 2013, 2009,...