Loudoun County Public Schools / Overview



Direct Variation

Two variables x and y show a __________________________ provided y = ax and a[pic] 0. The non-zero number a is called the _______________ and y is said to vary directly with x.

[pic]

Examples: Tell whether the equation represents a direct variation. If so, identify the constant of variation.

a) y = 3x b) y = x + 3 c) y = - ½ x

d) x + y = 7 e) 2x + y = 0 f) [pic]

Direct Variation Graphs

The graph of a direct variation equation is a __________________ with a slope of ____ and a y – intercept of _______.

So the line passes through the __________.

Examples: Graph the direct variation equation.

a) y = ¾ x b) y = -x

Inverse Variation

Two variables, x and y, vary inversely if they are related as follows:

[pic]

The constant a is the constant of variation, and y is said to vary inversely with x.

Examples: Tell whether the equation represents an inverse variation. If so, identify the constant of variation.

a) [pic] b) xy = 10 c) [pic]

d) [pic][pic] e) y = [pic] f) [pic]

Graphs of Inverse Variation

The graph of an inverse variation model is a curve that DOES NOT cross an axis.

EX:

How to determine if a set of data represents a direct variation.

If a set of data represents a direct variation you should be able to write a function rule in the form y = ax. The ratio _________ should always equal _____. [pic]

Examples: Does the set of data represent a direct variation? If so write the direct variation equation.

a)

|x |2 |4 |6 |8 |

|y |4 |8 |10 |12 |

b)

|x |2 |3 |4 |5 |

|y |-6 |-9 |-12 |-15 |

Writing direct variation equations given one set of data.

Use the set of data ( , ) to find your constant of variation______. Write the direct variation equation.

a) x = 6, y = 9 b) x = 3, y = -12

c) The amount of money m you make babysitting varies directly with the number h of hours you babysit. If you make $24 babysitting for 2 hours, write the direct variation equation. How much would you make if you babysat for 4.5 hours?

How to determine if a set of data represents an inverse variation.

Since inverse variation has the form [pic]. A table has inverse variation if xy gives the same answer for each ordered pair. The product should always equal _________.

[pic]

Examples: Does the set of data represent an inverse variation? If so write the inverse variation equation.

a)

[pic]

b)

[pic]

Writing the inverse variation equations given one set of data.

Use the set of data ( , ) to find your constant of variation. Write the inverse variation equation.

a) x = 9, y = 36 b) x = -21, y = 14

c) When setting up for a party, the hostess noticed the number of bananas varied inversely as the number of watermelon on each table. One table had 4 bananas and 5 watermelons. If another table had 7 bananas, how many watermelons would be needed to keep the same proportions?

Homework: Inverse and Direct Variation

Tell whether the equation represents direct variation, inverse variation or neither.

1. y = -4x 2. xy = 7 3. x = 7y

4. [pic]= 4 5. 3x + 4y = 8 6. 2xy = 20

7. x = -8 8. [pic] 9. y = x + 7

For each problem find: a) the direct variation b) the inverse variation

10. If y = 21 when x = 10, find y when x = 4.

Direct: Inverse:

11. If y = 17.5 when x = 12, find y when x = 8.

Direct: Inverse:

12. To travel a fixed distance, the rate is inversely proportional to the time required. When the rate is 60 miles per hour, the time required is 4 hours. What time would be required for the same distance if the rate were increased to 80 miles per hour?

13. Tell if the graph is direct, inverse, or neither?

[pic] [pic] [pic]

__________________ __________________ __________________

19.

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17. A farmer pumps water from an irrigation well to water his field. The time it takes to water the field varies inversely with the rate at which the pump operates. It takes 20 hours to water the field when the pumping rate is 600 gallons per minute. If he adjusts the pump so that it pumps at a rate of 400 gallons per minute, how long will it take to water the field?

F 12.5 hours

G 15 hours

H 30 hours

J 40 hours

16. If y varies directly as x and the constant of variation is -2, which equation represents this relationship?

A y = -2x

B y = [pic]

C y = [pic]

D y = 2x

15. Which of these equations is a direct variation?

F y = -8

G y = -8x

H y = -8x + 1

J y = -8x – 1

14. Which equation represents an inverse variation?

A [pic] = [pic]

B [pic] = [pic]

C 2a + 3 = 4b + 3

D [pic] = 7

18. If m varies directly as p, and m = 5 when p = 7, what is the constant of variation?

A 35

B 12

C [pic]

D [pic][pic]

5

6

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