ALGEBRA READINESS

LESSON 8-6Warm Up Lesson 8-6 Warm-Up

ALGEBRA READINESS

LESSON 8-6Warm Up Lesson 8-6 Warm-Up

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What is a “direct variation”?

Direct Variation (sometimes called a direct proportion): a linear (forms a line on a graph) function in the form of y = kx, where k ≠ 0 and coefficient k is called the “constant of the variation” (a number that never changes). This means that the y varies, or “changes”, directly, or proprtionally, with changes in x.

Note: Since y = 0 when x = 0, all direct variations pass through the origin (0, 0)

Examples: y = ¾ x y = -½ x

“Slope and Direct Variation” (8-6)

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How do you write the direct variation equation and graph it if you are given two points the line passes through?

Notice that the equation for direct variation, y = kx (where k is the constant of variation), is similar to the equation of a line, y = mx + b (where m is the slope and b is the y-intercept). That’s because the slope and the constant of variation are the same thing. Therefore, if you know the slope of a line, which is also the constant of variation, you can write the direct variation equation and graph it.

Example: Find the slope of the line that contains (1, 1.5) and (2, 3).

Step 1: Use the two given points to find the slope of the line.

The direct variation is y = 1.5x Substitute 1.5 for k in y = kx.

Step 2: Create a function table usingy = 1.5x .

Step 3: Graph the (x, y) ordered pairs.

“Slope and Direct Variation” (8-6)

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The number of miles traveled by a car varies directly

with the number of gallons of gas used. Use the variation m =

27g, where g is gallons of gas and m is miles driven. Find the

total miles the car can travel on 4 gallons of gas.

Write the direct variation.m = 27g

= 27 • 4 Substitute the number of gallonsof gas for g.

= 108 Simplify.

The car can travel 108 miles on 4 gallons of gas.

Slope and Direct VariationLESSON 8-6

Additional Examples

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Amber earns $7 per hour. Graph the relationship between

the number of hours Amber works and the amount she earns. Find

the slope of the line and explain what it represents.

Make a table.352821147 0Output ($)

54321 0Input (h)

The slope represents the amount Amber earns each hour, which is $7.

Plot the points. Connect themwith a line.

Slope and Direct VariationLESSON 8-6

Additional Examples

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How do you tell whether each “data pair” (x and y) in a table is a direct variation?

Tip: You can write y = kx as k = y / x if you divide both sides by x. If each data pair (x y) equals k (in other words, the ratio of y to x is the same for each x and y pair), then the table represents a direct variation.

Example: Is the following table a direct variation?

No, the ratio of (in other words, the “k”) is not the same for all of the x and y data pairs.

“Slope and Direct Variation” (8-6)

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1 –2 = –0.5

–1 2

–2 4

= –0.5

= –0.5

yxx y

–2 1

2 –1

4 –2

2 –1 = –2

21

–4 2

= 2

= –2

yxx y

–1 2

1 2

2 –4

For the data in each table, use the ratio to tell whether y

varies directly with x. If it does, write an equation for the direct

variation.

Yes, the constant of variation is –0.5. The equation is y = –0.5x.

yx

No, the ratio is not the

same for each pair of data.

yx

a. b.

Slope and Direct VariationLESSON 8-6

Additional Examples

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1. Christine makes $8 per hour babysitting. The amount of money she earns varies directly with the number of hours she works. Use the direct variation e = 8h, where h is the number of hours worked and e is the amount of money earned. Find how much Christine will make if she works 14 hours in one week.

2. One kilogram is equivalent to about 2.2 pounds. Graph the relationship between pounds and kilograms. Find the slope of the line and explain what it represents.

$112

8.86.64.42.2Pounds

4321Kilograms

The slope is 2.2. It represents the number of pounds per kilogram.

Slope and Direct VariationLESSON 8-6

Lesson Quiz