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

A power function - is any function in the form of y = kxn, where k is nonzero and n is a positive number (1, 2, and 3). The linear equation graph at the right shows that as the x-value increases, so does the y-value increase. For instance, if x = 2, y = 4. If x = 6 (multiplied by 3), then y = 12 (also multiplied by 3).

Direct variation – y varies directly as x means that y = kx, where k is the constant of variation. (see any similarities to y = mx + b ?) Another way of writing this is k = y/x In other words: As x increases in value, y increase or As x decreases in value, y decrease.

This is a graph of direct variation This is a graph of direct variation. If the value of x is increased, then y increases as well. Both variables change in the same manner. If x decreases, so does the value of y. It is said that y varies directly as the value of x. A direct variation between 2 variables, y and x, is a relationship that is expressed as: y = kx Direct variation with a power where n = 1. Others are y =kx2 and y = kx3

The relationship between y and x that is expressed as y = kx, k is called the constant of proportionality. In most problems, the k value needs to be found using the first set of data given. Example: The power, P, of a gear varies directly with the radius, r, of a gear. Find the constant of proportionality if P = 300 when r = 50 Start with the formula P = kr Typical problem, try it

In a factory, the profit, P, varies directly with the inventory, I In a factory, the profit, P, varies directly with the inventory, I. If P = 100 when I = 20, find P when I = 50. Step 1 Set up the formula P = Ik Step 2 Find the missing constant, k, for the given data. 100 = (20)k, k = 5 Step 3 Use the formula and constant to find the missing value. P = (50)(5) P = 250 It will be necessary to use the “first” set of data to find the value for the constant, k. y = kx

Note: x increases 6, 7, 8 And y increases 12, 14, 16 What is the constant of variation of the table above? Since y = kx, we can say k = y/x therefore, 12/6 = k or k = 2 14/7 = k or k = 2 16/8 = k or k = 2 Note: k stays constant Another example y = 2x is the equation!!!

Note: x decreases, 30, 15, 9 And y decreases. 10, 5, 3 What is the constant of variation of the table above? Since y = kx, we can say k = y/x therefore: 30/10 = k or k = 3 15/5 = k or k = 3 9/3 = k or k = 3 Note: k stays constant. Another example y = 3x is the equation

Note: x decreases, -4, -16, -40 And y decreases. -1, -4, -10 What is the constant of variation of the table above? Since y = kx, we can say k = y/x therefore: -1/-4 = k or k = ¼ -4/-16 = k or k = ¼ -10/-40 = k or k = ¼ Note: k stays constant y = ¼x is the equation

Use direct variation to solve word problems A car uses 8 gallons of gasoline to travel 290 miles. How much gasoline will the car use to travel 400 miles? Step 1: find points in the table x y 8 290 ? 400 Step 3: use the equation to find the unknown. Step 2: find the constant variation and equation. 400 = 36.25x 36.25 36.25 or x = 11.03 k = y/x or k = 290/8 or 36.25 y = 36.25x

Direct variation and its graph y = mx + b, m = slope and b = y-intercept With direct variation the equation is y = kx Note: m = k or the constant and b = 0, therefore the graph will always go through… The ORIGIN!!!!!

Solve the following variation problems using the formula. y varies directly as x. If y = 75 when x = 10, find y when x = 16. Your distance, d, from lightning varies directly with the time, t, it takes you to hear thunder. If you hear thunder 10 seconds after you see lightning, you are about 2 miles from the lightning. How far are you away from the lightning if you hear thunder in 3 seconds. The distance, d, a cyclist travels varies directly with the time, t, it takes in hours. If a cyclist travels 40 km in 2 hours, how far will he have traveled in 6 hours. More problems

4) The amount of sales tax on a new car is directly proportional to the purchase price of the car. If a $25,000 car cost $1750 in sales tax, what is the purchase price of a new car which has a $3500 sales tax? Hint: sales tax = k(purchase price) 5) The cost of a house in Florida is directly proportional to the size of the house. If a 2850 ft2 house cost $182,400, then what is the cost of a 3640 ft2 house?

Other Power Functions y = kx2 and y = kx3

y = kx2 The distance required for a moving car to stop varies directly as the square of the car’s speed. Therefore, the formula (equation) d = ks2 represents the stopping distance for the auto. A car traveling 50 miles per hour has a stopping distance of 135 feet. What would be the stopping distance of an auto travel 70 miles per hour. d = ks2 d = ks2 135 = k(502) d = .054(702) k = .054 d = 264.6 ft

y = kx3 The volume of a cube varies directly as the cube of the side lengths. If the volume (V) of a cube is 36cm with side lengths (s) of 2cm, what is the volume of a cube with side lengths (s) of 5. V = ks3 V = 4.5(53) 36 = k(23) y = 562.5 cm k = 4.5