the slope formula lesson 2.8 core focus on linear equations

The Slope Formula

Lesson 2.8

Core Focus onLinear Equations

Draw a line through each pair of points. Draw a slope triangle and find the slope.

1. (1, 3) and (5, 4)

2. (−2, 6) and (1, 0)

Warm-Up

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The Slope Formula

Find the slope of a line using the slope formula.

Lesson 2.8

Explore! Find That Formula

Step 1 Calculate the rate of change (the change in y over the change in x) for the table above.

Ginger got a job in downtown Oklahoma City. She boughta parking pass at a garage not far from her place of work.The table shows her total parking expenses based on thenumber of weeks she has been parking at the garage.

Step 2 Graph the ordered pairs on a Quadrant I coordinate plane like the one shown below. Draw a line through the points.

Step 3 Make a slope triangle and determine the slope of the line.

Step 4 What do you notice about the rate of change and the slope of the line?

Explore! Multi-Step Equation

Step 5 If you were given the table of values in the table to the right, what would the rate of change (or slope) ratio look like?

Step 6 The ratio developed in Step 5 is called the “SlopeFormula”. The subscripts identify two differentpoints. Try your formula on these points from thetable above: (6, 50) and (12, 86).

Did you get the same slope as you did in Steps 1 and 3?

Step 7 You have learned three methods for finding slope: rate of change, slope triangles and the slope formula. Which method do you like the best? Why?

The Slope Formula

The formula for the slope of a line that goes through a point with coordinates (x1, y1) and another point with coordinates (x2, y2) is:

2 1

2 1

Slopey y

x x

You read x1 as “x sub one.” Think of it as saying “the x-coordinate of the first point.”

Use the slope formula to find the slope of each line that passes through the given points.

a. (3, 2) and (8, 5)

Let (3, 2) be (x1, y1) and (8, 5) be (x2, y2).

Substitute the numbers intothe slope formula.

2 1

2 1

5 2 38 3 5

y y

x x

Example 1

Sometimes it helps to write the subscript letters over your

points to stay organized:x1 y1 x2 y2

(3, 2) (8, 5)


b. (1, –1) and (3, –5)

Let (1, –1) be (x1, y1) and (3, –5) be (x2, y2).


2 1

2 1

5 ( 1) 42

3 1 2

y y

x x

Example 1 Continued…


c. (6, –2) and (6, 4)

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


2 1

2 1

y y

x x

Example 1 Continued…

It is impossible to divide by 0 so the slope is undefined.

4 ( 2) 6undefined

6 6 0

Remember!

You have learned three methods for calculating slope. Allthree methods will work in any situation. Depending on theway the linear relationship is presented, there may be one methodthat is easier to use than the other two methods.

RATE OF CHANGEEasiest method when information is presented in an input-output table.

SLOPE TRIANGLE Easiest method when information is presented in a graph.

SLOPE FORMULA Easiest method when given two ordered pairs.

You can find slope from:• A table by finding rate of change.• A graph using a slope triangle.• The slope formula.

Which way do you prefer to find slope? Why?

Communication Prompt

Find the slope of each line that passes through the given points.

1. (4, 0) and (7, 9)

2. (9, 2) and (6, 4)

3. (4, −1) and (4, 3)

Exit Problems

3

Undefined

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the slope formula lesson 2.8 core focus on linear equations

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