DIRECT VARIATION

TO WRITE AND GRAPH AN EQUATION OF A DIRECT VARIATION

SIMPLIFY: (2)(-3)

FIND THE TENTH TERM OF THE SEQUENCE:

A(N) = 3 + (N – 1)(2)

Let the domain of the function f(x) = be {-5, 0, 10}. What is the range?

Find the equation of a line thatpasses through the points (-3, 4) and (2, -1).

Look carefully at the problem below.

A low-flow showerhead uses 5 gallons of water every 2 minutes. How many gallons would it use in 15 minutes?

Find a solution to the problem. Be prepared to explain how you arrived at your solution.

Let’s look at all the different strategies you used to solve the problem.

How would you represent this situation in a graph?• What are the variables involved?• What is the independent variable?• What is the dependent variable?• What amount of water would be used for zero minutes?• Is there a constant slope?• What does the slope represent in the problem?• Write an equation for the situation?• What is the slope?• What is the y-intercept?• What do the slope and y-intercept represent in the problem?

You have just investigated a direct variation, which is a relationship in which two variables are directionally proportional. The ratio between the two variables in a direct variation is constant.

A direct variation relationship can be represented with an equation of the form y = kx, where k is the constant of variation.

Since a direct variation is a proportional relationship, using an equation is an alternative for solving a proportion.

DIRECT VARIATION

Decide if the following scenarios are examples of direct variation. Justify your answer.

1.The amount of time spent running at a steady pace and the miles run.

2.Your age and a certain number of years from now.3.Your age and height.4.The amount of lawns you mow and the amount you make.5.The amount of gas you buy and the money you pay.6.The distance travelled and the speed of a car.7.The area of a circle and its radius.8.The circumference of a circle and its diameter.9.The age of Sam and his brother who is 5 years younger.

IDENTIFYING A DIRECT VARIATION

Does the equation represent a direct variation? If so, find the constant of variation.a. 7y = 2x

y = The equation has the form y = kx, so the equation is a

direct variation. Its constant of variation is .

b. 3y + 4x = 8 3y = -4x + 8

y = x + You cannot write the equation in the form y = kx. It is not

direct variation.

GOT IT?

Does 4x + 5y = 0 represent a direct variation? If so, find the constant variation.

Determine if -4 + 7x + 4 = 3y represents a direct variation. If it does, find the constant variation.

WRITING A DIRECT VARIATION EQUATION

Suppose y varies directly with x, and y = 35 when x = 5. What direct variation equation relates x and y? What is the value of y when x = 9?

1. Start with the function form of direct variation……y = kx2. Substitute 5 for x and 35 for y……..35 = k(5)3. Divide each side by 5 to solve for k…….7 = k4. Write an equation. Substitute 7 for k in y = kx……y = 7x5. The equation y = 7x relates x and y. 6. When x = 9……y = 7(9) = 63

GOT IT?

Suppose y varies directly with x, and y = 10 when x = -2. What direct variation equation relates x and y? What is the value of y when x = - 15?

Suppose y varies directly with x. Write a direct variation equation that relates x and y. Then find the value of y when x = 12. y = -10 and x = 2

A recipe for 12 muffins calls for 1 cup of flour. The number of muffins you can make varies directly with the amount of flour you use. You have 2½ cups of flour. How many muffins can you make?

The price p you pay varies directly with the number of pencils you buy. Suppose you buy 3 pencils for $0.51. How much is each pencil, in dollars?

GRAPHING A DIRECT VARIATION

Weight on Mars y varies directly with weight on Earth x. The weights of the science instruments onboard the Phoenix Mars Lander on Earth and Mars are: Mars = 50 lbs, Earth = 130 lbs.

What is an equation that relates weight, in pounds, on Earth x and weight on Mars y? y = kx 50 = k(130)

0.38 = k y = 0.38xThe equation y = 0.38x gives the weight y on Mars, in pounds, of an object that weighs x pounds on Earth.

Make a table and then draw a graph.

CONCEPT SUMMARY

The graph of a direct variation equation y = kx is a line with the following properties:

1.The line passes through (0,0).2.The slope of the line is k.

Time to reflect…….

1. How can you tell whether a situation represents a direct variation?

2. What does an equation for direct variation look like?

3. What does a graph for direct variation look like?

4. Give your own example of direct variation and explain why your example meets the criteria for direct variation.