A4.a How Do I Interpret Slope of a Line As A Rate Of Change? Course 3 Warm Up Warm Up Lesson...

A4.a How Do I Interpret Slope of a Line As A Rate Of Change?

Course 3

Warm UpWarm Up

Lesson PresentationLesson Presentation

Warm UpEvaluate each equation for x = –1, 0, and 1.

1. y = 3x2. y = x – 73. y = 2x + 54. y = 6x – 2

–3, 0, 3–8, –7, –63, 5, 7

Course 3

12-2 Slope of a Line

–8, –2, 4

Learn to find the slope of a line and use slope to understand and draw graphs.

Course 3


Linear equations have constant slope (constant rate of change). For a line on the coordinate plane, slope is the following ratio:vertical change

horizontal changechange in y change in x=

This ratio is often referred to as , or “rise over run,” where rise indicates the number of units moved up or down (vertical) and run (horizontal) indicates the number of units moved to the left or right. Slope can be positive, negative, zero, or undefined. A line with positive slope goes up from left to right. A line with negative slope goes down from left to right.

rise run

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You can use any two points to find the slope of the line.

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Helpful Hint

If you know any two points on a line, or two solutions of a linear equation, you can find the slope of the line without graphing. The slope of a line through the points (x1, y1) and (x2, y2) is as follows:

yy22 –– yy11 xx22 –– xx11

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Find the slope of the line that passes through (–2, –3) and (4, 6).

Example 1: Finding Slope, Given Two Points

Let (x1, y1) be (–2, –3) and (x2, y2) be (4, 6).

6 – (–3)4 – (–2)

Substitute 6 for y2, –3 for y1, 4 for x2, and –2 for x1.9

6=

The slope of the line that passes through (–2, –3) and (4, 6) is . 3

2

=y2 – y1x2 – x1

32=

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Find the slope of the line that passes through (–4, –6) and (2, 3).

Check It Out: Example 1

Let (x1, y1) be (–4, –6) and (x2, y2) be (2, 3).

3 – (–6)2 – (–4)

Substitute 3 for y2, –6 for y1, 2 for x2, and –4 for x1.9

6=

The slope of the line that passes through (–4, –6) and (2, 3) is . 3

2

=y2 – y1x2 – x1

32=

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Nonlinear equations have variable rates of change. This means that the rate of change is different between values. This is shown in a graph by a curved line.

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Determine whether each graph shows a constant or variable rate of change. Explain your reasoning.

Example 2A: Identifying Constant and Variable Rates of Change in Graphs

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The graph shows a constant rate of change. The slope between any two points is always the same.


Example 2B: Identifying Constant and Variable Rates of Change in Graphs

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The graph shows a variable rate of change. The slope between any two sets of points in Quadrant 1 is different.


Check It Out: Example 2A

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The graph shows a constant rate of change. The slope between any two points is always the same.


Check It Out: Example 2B

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The graph shows a variable rate of change. The slope is steeper at the ends than in the middle.

Example 3: Money Application

The table shows the total cost of fruit per pound purchased at the grocery store. Use the data to make a graph. Find the slope of the line and explain what it shows.

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Graph the data.

Pounds

Cost

Cost of Fruit

Additional Example 3 Continued

Find the slope of the line:

The slope of the line is 3. This means that for every pound of fruit, you will pay another $3.

=y3 – y2x3 – x2

15 5=

30 15 10 5

= 3

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Check It Out: Example 3The table shows the total cost of gas per gallon. Use the data to make a graph. Find the slope of the line and explain what it shows.

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Graph the data.Cost of Gas

Gallons Cost0 03 66 12

6

9

9

12

6 0

3

3 x

y

Gallons

Cost of Gas

Cost

Check It Out: Example 3 Continued

Find the slope of the line:

The slope of the line is 2. This means that for every gallon of gas, you will pay another $2.

=y3 – y2x3 – x2

6 3=

12 6 6 3

= 2

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Lesson Quiz: Part I

Find the slope of the line passing through each pair of points.

1. (4, 3) and (–1, 1)

2. (–1, 5) and (4, 2)

Insert Lesson Title Here

25

53–

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Lesson Quiz: Part II3. The table shows how much money Susan earned as a house painter for one afternoon. Use the data to make a graph. Find the slope of the line and explain what it shows.

Insert Lesson Title Here

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x

y

64 2 8 10 12 14 0

1020304050607080

The slope of the line is 7. This means Susan earned $7 for each hour worked.

A4.a How Do I Interpret Slope of a Line As A Rate Of Change? Course 3 Warm Up Warm Up Lesson...

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