Direct Variation Learn to recognize direct variation and identify the constant of proportionality.

Spiders How many legs does a spider have? 8 legs Therefore 2 spiders have a total of 16 legs 3 spiders have a total of 24 legs 4 spiders have a total of 32 legs And so on... This type of relationship is called...

Direct Variation The relationship between the amount of spiders and how many legs they have can be said to vary directly! We will be learning about equations, tables, and graphs of direct variations. Sometimes this is called direct proportion rather than direct variation but it is the same thing.

Direct Variation A direct variation relationship can be represented by a linear equation in the form y = kx, where k is a positive number called the constant of proportionality. The constant of proportionality can sometimes be referred to as the constant of variation.

y = kx When two variable quantities have a constant (unchanged) ratio, their relationship is called a direct proportion. We say, “y varies directly as x.” k is the constant of proportionality which means it never changes within a problem.

Finding Values Using a Direct Variation Equation At a frog jumping contest, Edward’s frog jumped 60 inches. Bella’s frog jumped 72 inches. Jacob’s frog jumped 6.5 feet. Use the equation y = 12x, where y is the number of inches and x is the number of feet to find the missing values of the table. Edward’s FrogBella’s FrogJacob’s Frog Inches (y)6072___ Feet (x)___66.5

Using Equations and Tables Edward’s FrogBella’s FrogJacob’s Frog Inches (y)6072___ Feet (x)___66.5 Substitute Solve Simplify 78 5

Now You Try Identify the constant of proportionality: 1. y = 15x y =.72x y = ¼x ¼

How about solving for k? When we are given the x and y values, we can solve for the constant of proportionality. Example If y varies directly with x, and y = 8 when x = 12, find k and write an equation that expresses this variation. Steps: 8 = k × 12 Substitute numbers into y = kx 8/12 = (k × 12)/12 Divide both sides by 12 2/3 = k Simplify y = 2/3x Plug k back into the equation

Now You Try If y varies directly as x, and x = 12 when y = 9, what is the equation that describes this direct variation? y = kx 9 = k × 12 9/12 = k ¾ = k y = ¾x

Now You Try If y varies directly as x, and x = 5 when y = 10, what is the equation that describes this direct variation? y = kx 10 = k × 5 10/5 = k 2 = k y = 2x

Copy and fill out tables: xy xy y = xy = 4x

Copy and fill out tables: xy xy y = 10x y = ½x ½ 1 3/2 2 5/2

Direct Variation Equation Sometimes we will have to put an equation into y = kx form and solve for y. Then we will be asked to identify k, the constant of proportionality. Examples: Tell whether each equation or relationship is a direct variation. If so identify the constant of proportionality. 1. 4y = 2x 2. ½y – ¾x = y – 5 = 3x

More Examples If y varies directly as x and y = 24 when x = 16, find y when x = 12. Solution: Set up a proportion since the ratios of corresponding values of x to y are always the same.