What is it and how do I know when I see it? Direct Variation What is it and how do I know when I see it?

Definition Direct Variation – a special type of linear relationship that can represented by a function in the form y = kx, Constant of Variation – is k, the coefficient of x, in the function y = kx.

Direct Variation A recipe for paella calls for 1 cup of rice to make 5 servings. In other words, a chef needs 1 cup of rice for every 5 servings. The equation y = 5x describes this relationship. In this relationship, the number of servings varies directly with the number of cups of rice.

Determine Direct Variation from a Table What happens if you solve y = kx for k? y = kx Divide both sides by x (x ≠ 0). So, in a direct variation, the ratio is equal to the constant of variation.

Example: Determine Direct Variation from a Table Tell whether the relationship is a direct variation. Explain. Find for each ordered pair. This is a direct variation because is the same for each ordered pair.

Examples of Direct Variation: What is the constant of variation of the table above? Since y = kx we can say Therefore: 12/6=k or k = 2 14/7=k or k = 2 16/8=k or k =2 Note k stays constant. y = 2x is the equation!

Examples of Direct Variation: What is the constant of variation of the table above? Since y = kx we can say Therefore: 30/10=k or k = 3 15/5=k or k = 3 9/3=k or k =3 Note k stays constant. y = 3x is the equation!

What is the constant of variation for the following direct variation? 2 -2 -½ ½ Answer Now

The k values are different! Is this a direct variation? If yes, give the constant of variation (k) and the equation. No! The k values are different!

Which is the equation that describes the following table of values? y = -2x y = 2x y = ½ x xy = 200 Answer Now

In a table, simplify any one of the ratios. We can find the constant of proportionality from a table of values, equation and a graph. In a table, simplify any one of the ratios. Chaperones 1 2 3 4 5 Students 12 24 36 48 60

Find the constant of proportionality: Example: Find the constant of proportionality: Apples (lbs) 2 2.5 3 3.5 4 Cost ($) 3.96 4.95 5.94 6.93 7.92

Using Direct Variation to find unknowns (y = kx) Given that y varies directly with x, and y = 28 when x=7, Find x when y = 52. HOW??? 2 step process 1. Find the constant variation k = y/x or k = 28/7 = 4 k=4 2. Use y = kx. Find the unknown (x). 52= 4x or 52/4 = x x= 13 Therefore: X =13 when Y=52

In a table, simplify any one of the ratios. We can find the constant of proportionality from a table of values, equation and a graph. In a table, simplify any one of the ratios. Chaperones 1 2 3 4 5 Students 12 24 36 48 60

Find the constant of proportionality: Apples (lbs) 2 2.5 3 3.5 4 Cost ($) 3.96 4.95 5.94 6.93 7.92 Click

Find the constant of proportionality: X Y 3 4.5 4 6 5 7.5 8 12 9 13.5 Click

Joke Time How would you describe a frog with a broken leg? Unhoppy What did the horse say when he got to the bottom of his feed bag? That’s the last straw! What kind of music do chiropractors listen to? Hip - Pop

the ORIGIN!!!!!

A Graph is a direct variation if: It is a straight line that passes through the origin (0,0)

Tell if the following graph is a Direct Variation or not. Yes No No No

Tell if the following graph is a Direct Variation or not. Yes No No Yes

Is the relationship shown in the graph proportional? 41 Is the relationship shown in the graph proportional? Yes No Hours Salary ($) 0 1 2 3 4 5 6 7 8 9 10 5 10 15 20 25 30 35 40 45 50 Answer: Yes

Is the relationship shown in the graph proportional? 42 Is the relationship shown in the graph proportional? Yes No 50 45 Cost ($) 40 35 30 25 20 15 Answer: No 10 5 0 1 2 3 4 5 6 7 8 9 10 Toppings

Is the relationship shown in the graph proportional? 43 Is the relationship shown in the graph proportional? Yes No Feet Seconds 0 1 2 3 4 5 6 7 8 9 10 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Answer: Yes

Is the relationship shown in the graph proportional? 44 Is the relationship shown in the graph proportional? Yes No Text Messages Cost ($) 0 1 2 3 4 5 6 7 8 9 10 5 10 15 20 25 30 35 40 45 50 Answer: No

Constant of Proportionality The graph of a proportional relationship is a straight line that passes through the origin. Proportional quantities can be described by the equation y= kx, where k is a constant ratio.

Constant of Proportionality Equation for constant of proportionality is y = kx We can tell that the relationship is directly proportional by looking at the graph. The graph is a straight line that passes through the origin.

Constant of Proportionality Equation for constant of proportionality is y = kx Create a table using the points from the graph: Total price (y) 20 40 60 80 100 Total pounds (x) 2 4 6 8 10 Divide total price by total pounds

Constant of Proportionality Equation for constant of proportionality is y = kx Create a table using the points from the graph: Total price (y) 20 40 60 80 100 Total pounds (x) 2 4 6 8 10 Divide total price by total pounds

In a graph, choose a point (x, y) to find and simplify the ratio. (2,24) Chaperones Students 0 1 2 3 4 5 6 7 8 9 10 6 12 18 24 30 36 42 48 54 60