slope of a line section 1.3. lehmann, intermediate algebra, 4ed section 1.3slide 2 introduction two...

Slope of a Line

Section 1.3

Lehmann, Intermediate Algebra, 4edSection 1.3 Slide 2

Introduction

Two ladders leaning against a building. Which is steeper?

We compare the vertical distance from the base of the building to the ladder’s top with the horizontal distance from the ladder’s foot to the building.

Comparing the Steepness of Two Objects

Lehmann, Intermediate Algebra, 4edSection 1.3 Slide 3

Introduction

Ratio of vertical distance to the horizontal distance:

Latter A:

Latter B:

So, Latter B is steeper.

8 feet 42 feet 1

8 feet 24 feet 1


Lehmann, Intermediate Algebra, 4edSection 1.3

To compare the steepness of two objects such as two ramps, two roofs, or two ski slopes, compute the ratio

for each object. The object with the larger ratio is the steeper object.

Slide 4

Property of Comparing the Steepness of Two Objects

vertical distancehorizontal distance

Property



Road A climbs steadily for 135 feet over a horizontal distance of 3900 feet. Road B climbs steadily for 120 feet over a horizontal distance of 3175 feet. Which road is steeper? Explain.

•These figures are of the two roads, however they are not to scale

Slide 5

Comparing the Steepness of Two Roads

Example

Solution



A: = = ≈

B: = = ≈

Slide 6



135 feet3900 feet

0.0351


120 feet3175 feet

0.0381

• Road B is a little steeper than road A

Solution Continued



The grade of a road is the ratio of the vertical to the horizontal distance written as a percent.

What is the grade of roads A?

Ratio of vertical distance to horizontal distance is for road A is 0.038 = 0.038(100%) = 3.8%.

Slide 7


Definition

Solution

Example

Finding a Line’s Slope


Let’s use subscript 1 to label x1 and y1 as the coordinates of the first point, (x1, y1). And x2 and y2 for the second point, (x2, y2).

Run: Horizontal Change = x2 – x1

Rise: Vertical Change = y2 – y1

The slope is the ratio of the rise to the run.Slide 8

Slope of a Non-vertical Line

We will now calculate the steepness of a non-vertical line given two points on the line.

Pronounced x sub 1 and y sub 1

Pronounced x sub 1 and y sub 1



Let (x1, y1) and (x2, y2) be two distinct point of a non-vertical line. The slope of the line is

Slide 9


vertical changehorizontal change

riserun

y2 – y1

x2 – x1 m = = =

In words: The slope of a non-vertical line is equal to the ratio of the rise to the run in going from one point on the line to another point on the line.

Definition



A formula is an equation that contains two or more variables. We will refer to the equation a

Slide 10


2 1

2 1

y ym

x x

as the slope formula.

Sign of rise or run

run is positive run is negativerise is positiverise is negative

Direction (verbal)

goes to the rightgoes to the left

goes upgoes down

(graphical)

Definition



Find the slope of the line that contains the points (1, 2) and (5, 4).

(x1, y1) = (1, 2)

(x2, y2) = (5, 4).

Slide 11

Finding the Slope of a Line

4 2 2 15 1 4 2

m

Example

Solution



A common error is to substitute the slope formula incorrectly:

Slide 12


Correct Incorrect Incorrect

2 1 2 1 2 1

2 1 1 2 2 1

y y y y x xm m m

x x x x y y

Warning




Slide 13


rise 2 2run 3 3

m

By plotting points, the run is 3 and the rise is –2.

Example

Solution



Increasing: Positive Slope Decreasing: Negative Slope

Slide 14

Definition

Positive risePositive run

m =

= Positive slope

negative risepositive run

m =

= negative slope

Increasing and Decreasing Lines


Find the slope of the line that contains the points (– 9 , –4) and (12, –8).

Slide 15


8 4 8 4 4 412 9 12 9 21 21

m

•The slope is negative

•The line is decreasing

–

Example

Solution



Find the slope of the two lines sketched on the right.

Slide 16

Comparing the Slopes of Two Lines

For line l1 the run is 1 and the rise is 2.

rise 12

run 2m

Example

Solution



Note that the slope of l2 is greater than the slope of l1, which is what we expected because line l2 looks steeper than line l1.

Slide 17

Comparing the Slopes of Two Lines

rise 44

run 1m

For line l2 the run is 1 and the rise is 4.

Solution Continued




Slide 18

Investigating Slope of a Horizontal Line

Plotting the points (above) and calculating the slope we get 3 3 0

06 2 4

m

The slope of the horizontal line is zero, no steepness.

Example

Solution

Horizontal and Vertical Lines



Slide 19

Investigating the slope of a Vertical Line

Plotting the points (above) and calculating the slope we get

5 2 3, division by zero is undefined.

4 4 0m

The slope of the vertical line is undefined.

Example

Solution



• A horizontal line has slope of zero (left figure).

• A vertical line has undefined slope (right figure).

Slide 20

Property

Property



Two lines are called parallel if they do not intersect.

Slide 21

Finding Slopes of Parallel Lines

Find the slopes of the lines l1 and l2 sketch to the right.

Definition

Example

Parallel and Perpendicular Lines


• Both lines the run is 3, the rise is 1

• The slope is,

Slide 22

Finding Slopes of Parallel Lines

• It makes sense that the nonvertical parallel lines have equal slope

• Since they have the same steepness

rise 1run 3

m

Solution

Parallel and Perpendicular Lines

slope of a line section 1.3. lehmann, intermediate algebra, 4ed section 1.3slide 2 introduction two...

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