Meaning of Slope for Equations, Graphs, and Tables

Meaning of Slope for Equations, Graphs, and Tables

Section 1.4

Lehmann, Intermediate Algebra, 3edSection 1.4

Find the slope of the line

Slide 2

Finding Slope from a Linear Equation

2 1.y x

x y

0 1

1 3

2 5

3 7

Create a table using x = 1, 2, 3. Then sketch the graph.

rise 22

run 1m

Example

Solution



Note the following three observations about the slope of the line

Slide 3


2 1.y x 1. The coefficient of x is 2, which is

the slope.

2. If the run is 1, then the rise is 2.

3. As the value of x increases by 1, the value of y increases by 2.

Observations



Find the slope of the line

Slide 4


3 8.y x

x y

0 8

1 5

2 2

3 –1

Create a table using x = 1, 2, 3. Then sketch the graph.

rise 33

run 1m

Example

Solution



For a linear equation of the form , m is the slope of the line.

Slide 5


y mx b

Are the lines parallel,

perpendicular, or neither?

53 and 12 10 5

6y x y x

Property

Example



• For the line the slope is

• For the other equation we solve for y:

Slide 6


53

6y x 5

6

12 10 5

12 10 10 5 10

12 10 5

12 10 512 12 12

5 56 12

y x

y x x x

y x

yx

y x

Original EquationAdd 10x to both sides.Combine & rearrange terms

Divide both sides by 12.

Simplify.

Property



• For the line the slope is

• Since the slopes are the same for both equations, the lines are parallel

Slide 7


5 56 12

y x 5

12

We use ZStandard followed by ZSquare to draw the line in the same coordinate system.

Solution Continued

Graphing Calculator



For the line , if the run is 1, then the rise is m.

Slide 8

Vertical Change Property

y mx b

Vertical Change property for a negative slope.

Vertical Change property for a positive slope.

Property

Vertical Change Property


Sketching Equations: • It’s helpful to know the y-intercept.• y-intercept has a x-value of 0.• Substitute x = 0 gives

Slide 9

Finding the y-intercept of a Linear Line

0y m b b

For a linear equation of the form , the y-intercept is (0, b).

y mx b Property

Finding the y-Intercept of linear Equation


What is the y-intercept of

Slide 10

Finding the y-intercept of a Linear Line

53?

6y x

• b is equal to 3, so the y-intercept is (0, 3)

If an equation of the form , we say that it is in slope-intercept form.

y mx b

Example

Solution

Definition

Finding the y-Intercept of linear Equation

Lehmann, Intermediate Algebra, 3edSection 1.4 Slide 11

Graphing Linear Equations

Sketch the graph of y = 3x – 1.

• The y-intercept is (0, –1) and the slope is 3 rise

31 run

To graph:

1.Plot the y-intercept, (0, 1). (continued)

Example

Solution




2. From (0, –1), look 1 unit to the right and 3 units up to plot a second point, which we see by inspection is (1, 2).

3. Sketch the line that contains these two points.

Solution Continued




To sketch the graph of a linear equation of the form

1.Plot the y-intercept (0, b).

2.Use m = to plot a second point.

3.Sketch the line that passes through the two plotted points.

riserun

y mx b

Guidelines




Sketch the graph of 2x + 3y = 6.

First we rewrite into slope-intercept form:2 3 6

2 3 2 6 2

3 2 6

3 2 63 3 3

x y

x y x x

y x

yx

Original EquationSubtract 2x from both sides.Combine & rearrange terms

Divide both sides by 3.

Example

Solution




y-intercept: (0, 2) Slope:

1. Plot the y-intercept, (0, 2).

2. From the point (0, 2), look 3 units to the right and 2 units down to plot a second point, which we see by inspection is (3, 0).

2 2 rise3 3 run

22

3y x -a a

Simplfy : =-b b

Solution Continued




3. Then sketch the line that contains these two points. We can verify our result by checking that both (0, 2) and (3, 0) are solutions.

Solution Continued




1. Determine the slope and the y-intercept of ax + by =c, where a, b, and c are constants and b is nonzero.

2. Find the slope and the y-intercept of the graph of 3x + 7y = 5.

First we rewrite into slope-intercept form:

Example

Solution




Slope is and the y-intercept is

ax by c

ax by ax c ax

by ax c

by a cx

b b ba c

y xb b

Divide both sides by b.

-a aSimplfy : =-

b b

Original equation

Subtract ax from both sides.

Combine and rearrange terms.

0, .cb

ab

Solution Continued




Given that ax + by = c in slope-intercept form is .

3 c 5 and the y-intercept, 0, 0, .

7 b 7ba

a cy x

b b , then given 3x + 7y = 5, we substitute .

3 for a, 7 for b and 5 for c. Thus, the slope, .

Solution Continued



For the following sets, is there a line that passes through them? If so, find the slope of that line.

Slide 20

Slope Addition Property

• Value of x increases by 1.

• Value of y changes by –3.

•The slope is –3.

Example

Solution




Set 2 • Value of x increases by 1. • Value of y changes by 5.

So, the slope is 5.

Set 3• Value of x increases by 1. •Value of y does not change by the same value. Hence, not a line.

Solution Continued


Meaning of Slope for Equations, Graphs, and Tables

Documents

Transcript of Meaning of Slope for Equations, Graphs, and Tables