Lesson 5-2

Slope and Direct Variation

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Objectives

• Write and graph direct variation equations

• Solve problems involving direct variation

Vocabulary

• Constant of variation – the number k in equations the form y = kx

• Graph – is a plot of an ordered pair (x,y) on the coordinate plane

• Family of graphs – graphs and equations of graphs that have at least one characteristic in common

• Parent graph – the simplest of the graphs in a family of graphs

y

x

y

x

x-y Coordinate Plane

Slope, m (∆y / ∆x)Point Plotting

(x, y) (-4, 7) (5, -8)

x – left or righty – up or down

right 5

down 8

left 4

up 7

(x, y) (-4, 7) (5, -8)

rise: ∆y – change in y (7 - -8 = 15)run: ∆x – change in x (-4 – 5 = -9)

∆y = 15

∆x = -9

5m = ------ -3

rise

run

Example 1a

Name the constant ofvariation for the equation.Then find the slope of theline that passes through thepair of points.

Slope formula

Simplify.Answer: The constant of variation is 2. The slope is 2.

Example 1b

Name the constant ofvariation for the equation.Then find the slope of theline that passes through thepair of points.

Slope formula

Simplify.Answer: The constant of variation is –4. The slope is –4.

Example 2

Step 1 Write the slope as a ratio.

Step 2 Graph (0, 0).

Step 3 From the point (0, 0), move up 1 unit and right 1 unit. Draw a dot.

Step 4 Draw a line containing the points.

Example 3

Step 1 Write the slope as a ratio.

Step 2 Graph (0, 0).

Step 3 From the point (0, 0), move down 3 units andright 2 units. Draw a dot.

Step 4 Draw a line containing the points.

Example 4

Suppose y varies directly as x, and y = 9 when x = -3

Write a direct variation equation that relates x and y.Find the value of k.

Direct variation formula

Replace y with 9 and x with –3.

Divide each side by –3.

Simplify.

Answer: Therefore,

Example 5

Travel The Ramirez family is driving cross-country on vacation. They drive 330 miles in 5.5 hours.

Write a direct variation equation to find the distance driven for any number of hours.

Words The distance traveled is 330 miles, and the time is 5.5 hours.

Variables

Distance equals rate times time.

Equation 330 mi r 5.5h

Example 5 cont

Solve for the rate.

Answer: Therefore, the direct variation equation is

Original equation

Divide each side by 5.5.

Simplify.

Summary & Homework

• Summary:– A direct variation is described by an equation of

the form y = kx, where k ≠ 0– In y = kx, k is the constant of variation. It is also

the slope of the related graph

• Homework: – Pg 268 16-46 even