Chapter 3 Section 3. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. The Slope of...

Chapter 3 Section 3

Objectives

1

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

The Slope of a Line

Find the slope of a line, given two points.

Find the slope from the equation of a line.

Use slopes to determine whether two lines are parallel, perpendicular, or neither.

3.3

2

3


An important characteristic of the lines we graphed in Section 3.2 is their slant, or “steepness.”

One way to measure the steepness of a line is to compare the vertical change in the line with the horizontal change while moving along the line from one fixed point to another. This measure of steepness is called the slope of the line.

The Slope of a Line

Slide 3.3-3


Objective 1


Slide 3.3-4


To find the steepness, or slope, of the line in the figure below, begin at point Q and move to point P. The vertical change, or rise, is the change in the y-values, which is the difference 6 − 1 = 5 units. The horizontal change, or run, is the change in the x-values, which is the difference 5 − 2 = 3 units.

vertical change in ri

horizontal changeslope

in

se

run 3( )

5

x

ym

The slope is the ratio of the vertical change in y to the horizontal change in x.

Count squares on the grid to find the change. Upward and rightward movements are positive. Downward and leftward movements are negative.

Slide 3.3-5



Solution:

Find the slope of the line.

6

1m

6m

Slide 3.3-6

EXAMPLE 1 Finding the Slope of a Line


Slope FormulaThe slope m of a line through the points (x1, y1) and (x2, y2) is

The slope of a line can be found through two nonspecific points. This notation is called subscript notation, read x1 as “x-sub-one” and x2 as “x-sub-two”.

21 2

2 1

1

horizontal change in ru(where ).

vertical change in rise

n x x

x

y

x xm

y y

Moving along the line from the point (x1, y1) to the point (x2, y2), we see that y changes by y2 − y1 units. This is the vertical change (rise). Similarly, x changes by x2 − x1 units, which is the horizontal change (run). The slope of the line is the ratio of y2 − y1 to x2 − x1.

Slide 3.3-7

Find the slope of a line, given two points. (cont’d)

The slope of a line is the same for any two points on the line.


Solution:

4 8

2 6m

12

8

3

2

Find the slope of the line through (6, −8) and (−2, 4).

and yield the same slope. Make sure to start with the

x- and y-values of the same point and subtract the x- and y-values of the other point.

2 1

2 1x

y y

x 1 2

1 2x

y y

x

Slide 3.3-8

EXAMPLE 2 Finding Slopes of Lines


Orientation of Lines with Positive and Negative SlopesA line with a positive slope rises (slants up) from left to right.

A line with a negative slope falls (slants down) from left to right.

Slopes of Horizontal and Vertical LinesHorizontal lines, with equations of the form y = k, have slope 0.

Vertical lines, with equations of the form x = k, have undefined slopes.

Slide 3.3-9

Find the slope of a line, given two points. (cont’d)


Solution:

5 5

1 2m

0

3

0

Find the slope of the line through (2, 5) and (−1, 5).

Slide 3.3-10

EXAMPLE 3 Finding the Slope of a Horizontal Line


Solution:

3 3

4 1m

5

0

Find the slope of the line through (3, 1) and (3,−4).

undefined slope

Slide 3.3-11

EXAMPLE 4 Finding the Slope of a Vertical Line


Objective 2


Slide 3.3-12


3 5y x

7y

3 5y x

11y

Consider the equation y = −3x + 5.

The slope of the line can be found by choosing two different points for value x and then solving for the corresponding values of y. We choose x = −2 and x = 4.

3 52y 6 5y

3 4 5y 12 5y

The ordered pairs are (−2,11) and (4, −7). Now we use the slope formula.

11 7 18

63

2 4m

Slide 3.3-13



Step 1: Solve the equation for y.

Step 2: The slope is given by the coefficient of x.

The slope, −3 is found, which is the same number as the coefficient of x in the given equation y = −3x + 5. It can be shown that this always happens, as long as the equation is solved for y.

Finding the Slope of a Line from Its Equation

Slide 3.3-14

Find the slope from the equation of a line. (cont’d)


Solution:

33 32 9xyx x

Find the slope of the line 3x + 2y = 9.

3 9

2 2y x 3

2m

2 3

2 2

9y x

Slide 3.3-15

EXAMPLE 5 Finding Slopes from Equations


Objective 3


Slide 3.3-16


Two lines in a plane that never intersect are parallel. We use slopes to tell whether two lines are parallel. Nonvertical parallel lines always have equal slopes.


Lines are perpendicular if they

intersect at a 90° angle. The product

of the slopes of two perpendicular

lines, neither of which is vertical, is

always − 1. This means that the

slopes of perpendicular lines are

negative (or opposite) reciprocals — if

one slope is the nonzero number a,

the other is . The table to the right

shows several examples.

1

a

Slide 3.3-17


Slopes of Parallel and Perpendicular LinesTwo lines with the same slope are parallel.

Two lines whose slopes have a product of − 1 are perpendicular.

Slide 3.3-18

Use slopes to determine whether two lines are parallel, perpendicular, or neither. (cont’d)


Solution:

3 4x y 3 9x y

3 33 4x xyx 3 9xy xx

4 3

1 1

xy

4 3y x

3 9

3 3 3

xy

13

3y x

13 1

3

3m 1

3m

The product of their slopes is − 1, so they are perpendicular

Decide whether the pair of lines is parallel, perpendicular, or neither.

Slide 3.3-19

EXAMPLE 6 Deciding Whether Two Lines Are Parallel or Perpendicular

Chapter 3 Section 3. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. The Slope of...

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Transcript of Chapter 3 Section 3. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. The Slope of...