7.3 Direct Variation

Direct Variation

Direction variation is used in many every day life applications. One example is distance; d=rt is just one example of a direction variation.

Direct Variation defined

Constant of Variation

In the formula, y=kx, k is referred to as the constant of variation OR the constant of proportionality. It is the coefficient of x (not y!)

A constant does not change in value.

What does it take to be a direct variation?

• The line on a graph will ALWAYS pass through the origin!

• Nothing is added or subtracted from kx in the formula; there is no "b". It is simply y = kx (y = mx would also work)

• All ordered pairs are proportional. If you divide the y from one ordered pair by its x and do that for all ordered pairs, the ratios will be equal!

Multiple Select

Which of the following ARE direct variations based on the definition and the three requirements?

• 3x + 5y = 1

• -12x = 6y

Fill in the Blanks

2x + y = 0 is an example of a direct variation and the constant of variation is

(do the math!)

Multiple Select

Is -x +y = 1 an example of a direct variation?

• Yes

• No

How to graph a DIRECT VARIATION

Step 1. Plot a point on the origin, (0,0).

Step 2. Use k like you would slope to plot two more points.

Step 3. Connect the points with a line (with arrows on each end!)

Writing a DV from a graph

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The direct variation equation for the real life example shown on the graph is

Y varies directly with x

This video is less than two minutes in length and shows one way of solving "varies directly with x" problems. ("directly proportional problems are done the same way)

Open Ended

Why does C vary directly with s?

Open Ended

What is the direct variation equation that relates C to s?

How to determine a direct variation from a table

Remember that y = kx is the same as y = ax.

Constant of proportionality from tables

For each ordered pair given, (-4,1),(-3,0.75). (-2. 0.5) and (-1,0.25) if you divide the y value by the x value you find k for the ordered pairs. IF k is the same for each, then you have a direct variation and have found the constant of variation/the constant of proportionality

Multiple Select

Does the table illustrate a direct variation?

• No

• Yes

Direct Variation

Direct variation describes a simple relationship between two variables . We say y varies directly with x if: y=kx. for some constant k , called the constant of variation or constant of proportionality .

Direct Variation

The Graph of a Direct Variation will graph a straight line that will pass through the origin. The slope can be positive or negative. The slope of the line will equal the constant of variation.

Multiple Choice

Which of the given graphs is a direct variation?

• A

• B

• C

• D

Multiple Choice

Is this a direct variation?

• yes

• no

Multiple Choice

Direct Variation?

• yes

• no

Multiple Choice

Is the given table a direct variation? If so, what is the constant of variation?

(in a direct variation - if x increases, then y also increases - proportionally. If x decreases, then y also decreases - proportionally)

• Yes; 1/3

• Yes; 3

• Yes; 5

• No

Multiple Choice

Which graph shows a proportional relationship between x and y?

• A

• B

• C

• D

Direct Variation

Using Tables

Multiple Choice

Write the equation for the table given.

• y = 1/3x

• y = 3x

• y = 1/2x

Multiple Choice

Which of the following is NOT a direct variation?

• y = -2x

• y = 1/2x

• 2x + 3y = 2

• 3y = 4x

Multiple Choice

Is the given table a direct variation? If so, what is the constant?

• Yes; 1/4

• Yes; 4

• Yes; 2

• No

Multiple Choice

Write an equation for this relationship.

• y = 4x

• y = 1/8x

• y = 1/4x

• y = 8x

Direct Variation

You can solve problems involving direct variation. Just Remember

y/x = y/x..

Multiple Choice

On a treasure map, the actual distance is directly related to the scaled distance.  If 15 miles is represented by 3 centimeters, then how many centimeters on the treasure map are equivalent to 135 miles?

• 27 cm

• 675 cm

• 529 cm

• 32 cm

Multiple Choice

In a direct variation, y = 39 when x = 3. 
Find the value of y when x = 2.

• 26

• 3

• 58.5

• 2

Multiple Choice

If y varies directly with x and y= 12 when x= 15. What is the value of x when y= 16?

• 12.8

• 20

• 11.25

• 19

Multiple Choice

Trina's paycheck earnings varies directly as the number of hours she works. If she works 19 hours and earns $187.15, what should she earn if she worked 40 hours? 

• $394

• $440

• $472.80

• $512.20

Multiple Choice

Which of the given graphs is a direct variation?

• A

• B

• C

• D

Direct Variation

Using Tables

Multiple Choice

Write the equation for the table given.

• y = 1/3x
• y = 3x
• y = 1/2x

Multiple Choice

Which of the following is NOT a direct variation?

• y = -2x
• y = 1/2x
• 2x + 3y = 2
• 3y = 4x

Multiple Choice

Is the given table a direct variation? If so, what is the constant?

• Yes; 1/4
• Yes; 4
• Yes; 2
• No

Multiple Choice

Write an equation for this relationship.

• y = 4x
• y = 1/8x
• y = 1/4x
• y = 8x

What is direct variation?

• Direct variation describes a simple relationship between two variables . We say y varies directly with x (or as x , in some textbooks) if: y=kx. for some constant k , called the constant of variation or constant of proportionality 

Multiple Choice

On a treasure map, the actual distance is directly related to the scaled distance.  If 15 miles is represented by 3 centimeters, then how many centimeters on the treasure map are equivalent to 135 miles?

• 27 cm
• 675 cm
• 529 cm
• 32 cm

Examples of Direct Variation

• The number of hours you work DIRECTLY impacts the amount of money you make

• The amount of weight on a spring DIRECTLY impacts the distance the spring will stretch

• The speed of a car DIRECTLY impacts the distance traveled in a certain amount of time.

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y varies directly with x.  When y=3, x=9.  Find x when y=12.  

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y varies directly as x. When y=10, x=2. Find y when x=13.

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The weight of an object on Earth varies directly as the weight of the same object on the moon. A 300 pound object would weigh 48 pounds on the moon. How much would a 65 pound object weigh on the moon?

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The amount of money raised at a charity fundraiser is directly proportional to the number of attendees. The amount of money raised for five attendees was $100. How much money will be raised for 60 attendees?

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Fill in the missing value on the table

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Write and solve a direct variation equation to find the answer.

If y varies directly with x and y and y = 18 when x = 3, find y when x = 2.

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Write and solve a direct variation equation to find the answer.

If y varies directly with x and y and y = 20 when x = 10, find y when x = 4.

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Write and solve a direct variation equation to find the answer.

If y varies directly with x and y and y = 8 when x = 2, find y when x = 1.

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Write and solve a direct variation equation to find the answer.

If y varies directly with x and y and y = 20 when x = -2, find y when x = -1.

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Write and solve a direct variation equation to find the answer.

If y varies directly with x and y and y = 8 when x = 4, find y when x = 3.

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Write and solve a direct variation equation to find the answer.

If y varies directly with x and y and y = -6 when x = -3, find y when x = 1.

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Write and solve a direct variation equation to find the answer.

If y varies directly with x and y and y = 87 when x = 3, find x when y = 29.

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Write and solve a direct variation equation to find the answer.

A direct variation includes the points (2, -10) and (n, 5). Find n.

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Write and solve a direct variation equation to find the answer.

A direct variation includes the points (43, 86) and (n, 78). Find n.

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Write and solve a direct variation equation to find the answer.

A direct variation includes the points (4, 68) and (1, n). Find n.

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Write and solve a direct variation equation to find the answer.

A direct variation includes the points (-18, 18) and (2, n). Find n.