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LESSON 8-5Warm Up

Lesson 8-5 Warm-Up

ALGEBRA READINESS

LESSON 8-5Warm Up Lesson 8-5 Warm-Up

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“Slope” (8-5)What is “rate of change”

How can you find the “rate of change”?

rate of change: a unit rate that tells how fast the dependent variable (y) is changes in relation to the dependent variable (y) – In other words, “x changes _?_ much for every change of 1 in y”

Since the independent variable is plotted on the x-axis (the x’s) and the dependent variable is plotted on the y-axis (the y’s), we can find the rate of change by using the following rule:

Rule:

Since rate of change is a linear function (forms a line when graphed), you can use two point on the line to find the vertical and horizontal changes, such as (x1, y1) and (x2,, y2). To find the vertical change, find the difference in the y terms (y2,- y1), and to find the horizontal change, find the difference in the x terms (x2,- x1)

Example: The following graph shows the altitude of an airplane as it’s coming in for a landing. Find the rate of change.

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The rate of change is , which means the airplane descends

feet every second.

“Slope” (8-5)

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What is “slope”

How can you find the “rate of change”?

slope: the rate of change of a line on a graph in a unit rate form (how much the y changes for every 1 change in x)

Rule:

You only need two points on the line to find slope (just like you do for rate of change). You can either: 1. divide the difference of the y values by the difference of the x values or 2. simply divide the rise (vertical distance between the two points) by the run (horizontal distance between the two points) to find slope as in the following example:

Note: When using the subtraction method, make sure you start with the same point for the y’s on top and the x’s on the bottom.

Example: Find the slope of the line.

“Slope” (8-5)

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How do you determine whether a slope (the rate of change) is positive, negative, zero, or undefined (impossible)? can you tell if a slope is positive, negative, 0 (no slope), or undefined (impossible)?

“Slope” (8-5)

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slope =riserun

Find the slope of the line.

–42= Substitute the rise and run.

= – 2 Simplify.

The slope of the line is –2.

SlopeLESSON 8-5

Additional Examples

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Find the slope of the line that contains the points (1, 3)

and (–2, 2)

slope =change in y-coordinateschange in x-coordinates

= 3 – 21 – (–2)

Subtract the coordinates of the secondpoint from those of the first.

13= Simplify.

SlopeLESSON 8-5

Additional Examples

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slope =riserun

Find the slope of each line.

a.

4 – 1

0 – 2

=

= 3 –2

= – 32

The slope of the line is – .32

SlopeLESSON 8-5

Additional Examples

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The slope of the line is 2.

slope =riserun

b.

(continued)

= 2

–1 – 1

–2 – (–1)

=

= –2–1

SlopeLESSON 8-5

Additional Examples

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slope =y2 – y1

x2 – x1

The slope of EF is – . 15

Find the slope of the line through E(3, –2) and F(–2, –1).

Substitute (–2, –1) for (x2, y2) and(3, –2) for (x1, y1).

–1 – (–2) –2 – 3

=

= – 15= 1

–5Simplify.

SlopeLESSON 8-5

Additional Examples

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= 0

2 – 2

1 – (–4)

= Substitute (1, 2) for

(x2, y2) and (–4, 2)

for (x1, y1).

slope =y2 – y1

x2 – x1

The slope of the horizontal line is 0.

Find the slope of each line.

a.

= 0 5 Simplify.

SlopeLESSON 8-5

Additional Examples

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Division by zero is undefined.

The slope of the vertical line is undefined.

b.

(continued)

Substitute (2, 1) for (x2, y2) and (2, –4) for (x1, y1).

= 1 – (–4)

2 – 2

= 5 0

Simplify.

slope =y2 – y1

x2 – x1

SlopeLESSON 8-5

Additional Examples

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1. Find the slope of each line.

 

2. the line that passes through the points (1, 3) and (2, 6)

3. the line that passes through the points (–3, 4) and (3, –5)

2

3

32

–

SlopeLESSON 8-5

Lesson Quiz