2. Chapter 2: Variation and Graphs Vogler Algebra II

3. Variation The DRIP system is a handy way to think about variation: Direct variation = Ratios are equal Inverse variation = Products are equal The distance that you travel varies directly with your rate. If you travel 150 miles in 2 hours, how many miles will you travel in 5? Constant of variation

4. Variation Direct variation equations follow this pattern: y=kx where k is the constant of variation. The equation would be: y=75x. How many miles could be traveled in 3 hours? The important thing to remember is that in Direct variation the RATIO gives you the constant of variation.

5. Variation Braking distance is directly proportional to the square of a vehicle’s speed. If a car needs 25 ft. to stop at 20 MPH, how far will it need to stop at a speed of 60 MPH? The equation for braking distance would be: y=1/16 x 2. What is the braking distance for a car traveling at 15 MPH? Constant of variation

6. Variation Inverse variation = Products are equal The law of the lever states that the weight needed to balance an object is inversely proportional to the distance of the weight from the fulcrum (balance point). If Mr. Vogler weighs 235 lb. and is sitting 2 feet from the fulcrum, how far does his son (who weighs 4 lb) have to sit from the fulcrum to balance his morbidly obese father. 235 x 2 = 4s

7. Variation Inverse variation equations follow this pattern: y=k/x where k is the constant of variation. The lever equation would be y=470/x. How far away would a person have to sit who weighs 100 lb? Remember that in Inverse variation the PRODUCT gives you the constant of variation.

8. Variation Teenagers’ intelligence varies inversely as their age. If a student has an IQ of 200 at age 15, what is the IQ of a student when they are 18? 200 x 15 = 18a The intelligence equation would be y=3000/x. How smart is a 5th grader?

9. Variation: Fundamental Theorem The Fundamental Theorem of Variation governs the relationship between changes in x (independent variable) and y (dependent variable): Direct variation: if x n is multiplied by c, then y is multiplied by c n. Y varies directly as the cube of x. What is y if x is doubled? 2 3 =8 y=4x 2. If x multiplied by 3, then what should y be?

10. Variation: Fundamental Theorem Inverse variation: if k/x n is multiplied by c, then y is multiplied by 1/c n. Y varies inversely as x. What is y if x is multiplied by 6? Find the reciprocal of 6 1 = 1/6. y=25/x 5. If x is multiplied by 7, what is y? What if x is multiplied by 1/3?

11. Graphs: linear functions y=mx X = input/independent variable Y= output/dependent variable Domain and range can be infinite Slope: rise/run

12. Modeling: Linear functions A child is riding his bicycle at a constant velocity. Give an equation to model the distance he covers each hour.

13. Graphs: quadratic functions y=ax 2 Domain: all real numbers Range: Depends on a If a>0, then all positive real numbers If a<0, then all negative real numbers Slope is constantly changing

14. Modeling: Quadratic functions A vehicle accelerates from a stop. Give an equation to model the distance covered during each second of its acceleration.

15. Graphs: hyperbolas y=k/x Y=k/x Domain and range are all reals except 0 Asymptotic to x and y axis Y=-k/x Domain and range are all reals except 0 Asymptotic to x and y axis

16. Graphs: hyperbolas y=k/x 2 Y=k/x 2 Domain is all reals except 0. Range is all positive reals except 0 Y=-k/x 2 Domain is all reals except 0. Range is all negative reals except 0

17. Modeling: hyperbolas A cat closes the distance between itself and a mouse. Give the equation that describes this situation.