1 / 16

The Power Functions

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 = kx n , where k is nonzero and n is a positive number (1, 2, and 3).

ide
Download Presentation

The Power Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. 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).

  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.

  4. This is a graph of directvariation. 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

  5. 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 = 50Start with the formulaP = kr Typical problem, try it

  6. In a factory, the profit, P, varies directly with the inventory, I. If P =100 when I = 20, find P when I = 50.Step 1Set up the formulaP = IkStep 2Find the missing constant, k, for the given data.100 = (20)k, k = 5Step 3Use 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

  7. 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 = 216/8 = k or k = 2 Note: k stays constant Note: x increases 6, 7, 8 And y increases 12, 14, 16 Another example y= 2x is the equation!!!

  8. 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 = 39/3 = k or k = 3 Note: k stays constant. Note: x decreases, 30, 15, 9 And y decreases. 10, 5, 3 Another example y= 3x is the equation

  9. 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 Note: x decreases, -4, -16, -40 And y decreases. -1, -4, -10 y = ¼x is the equation

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

  11. 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!!!!!

  12. 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

  13. 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?

  14. Other Power Functions y = kx2 and y = kx3

  15. y = kx2 The distance required for a moving car to stop variesdirectly as the square of the car’s speed. Therefore, the formula (equation) d = ks2represents 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

  16. 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

More Related