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Meaning of Slope for Equations, Graphs, and Tables

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Section 1.4

Meaning of Slope for Equations, Graphs, and Tables

Section 1.4

Slide 2

Finding Slope from a Linear Equation

Finding Slope from a Linear Equation

Example

Find the slope of the line

Solution

x y

0 1

3

5

3 7

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

Section 1.4

Slide 3

Finding Slope from a Linear Equation

Finding Slope from a Linear Equation

Observations

Note the following three observations about the slope of the line

The coefficient of x is 2, which is the slope.

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

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

Section 1.4

Slide 4

Finding Slope from a Linear Equation

Finding Slope from a Linear Equation

Example

Find the slope of the line

Solution

x y

0 8

5

2

3 –1

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

Section 1.4

Slide 5

Finding Slope from a Linear Equation

Finding Slope from a Linear Equation

Property

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

Example

Are the lines parallel,

perpendicular, or neither?

Section 1.4

Slide 6

Finding Slope from a Linear Equation

Finding Slope from a Linear Equation

Property

For the line the slope is

For the other equation we solve for y:

Original Equation

Add 10x to both sides.

Combine & rearrange terms

Divide both sides by 12.

Simplify.

Section 1.4

Slide 7

Finding Slope from a Linear Equation

Finding Slope from a Linear Equation

Solution Continued

For the line the slope is

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

Graphing Calculator

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

Section 1.4

Slide 8

Vertical Change Property

Vertical Change Property

Property

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

Vertical Change property for a positive slope.

Vertical Change property for a negative slope.

Section 1.4

Slide 9

Finding the y-intercept of a Linear Line

Finding the y-Intercept of linear Equation

Sketching Equations:

It’s helpful to know the y-intercept.

y-intercept has a x-value of 0.

Substitute x = 0 gives

Property

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

Section 1.4

Slide 10

Finding the y-intercept of a Linear Line

Finding the y-Intercept of linear Equation

Example

What is the y-intercept of

Solution

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

Definition

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

Section 1.4

Slide 11

Graphing Linear Equations

Graphing Linear Equations

Example

Sketch the graph of y = 3x – 1.

Solution

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

To graph:

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

Section 1.4

Slide 12

Graphing Linear Equations

Graphing Linear Equations

Solution Continued

- 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).
- Sketch the line that contains these two points.

Section 1.4

Slide 13

Graphing Linear Equations

Graphing Linear Equations

Guidelines

- To sketch the graph of a linear equation of the form
- Plot the y-intercept (0, b).
- Use m = to plot a second point.
- Sketch the line that passes through the two plotted points.

Section 1.4

Slide 14

Graphing Linear Equations

Graphing Linear Equations

Example

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

Solution

First we rewrite into slope-intercept form:

Original Equation

Subtract 2x from both sides.

Combine & rearrange terms

Divide both sides by 3.

Section 1.4

Slide 15

Graphing Linear Equations

Graphing Linear Equations

Solution Continued

- y-intercept: (0, 2) Slope:
- 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).

Section 1.4

Slide 16

Graphing Linear Equations

Graphing Linear Equations

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.

Section 1.4

Slide 17

Graphing Linear Equations

Graphing Linear Equations

Example

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.

Solution

First we rewrite into slope-intercept form:

Section 1.4

Slide 18

Graphing Linear Equations

Graphing Linear Equations

Solution Continued

Slope is and the y-intercept is

Section 1.4

Slide 19

Graphing Linear Equations

Graphing Linear Equations

Solution Continued

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

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

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

Section 1.4

Slide 20

Slope Addition Property

Slope Addition Property

Example

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

Solution

- Value of x increases by 1.
- Value of y changes by –3.
- The slope is –3.

Section 1.4

Slide 21

Slope Addition Property

Slope Addition Property

Solution Continued

- 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.