1. Slope. Direct Variation What you’ll learn To write and graph an equation of a direct variation. Direct variation and Constant of variation for a direct variation To find the slope using a graph without the formula.

2. Problem 1: Finding the slope. If you travel at 50 mph for one hour, then you would have traveled 50 miles. If you travel for 2 hours at that speed, you would have traveled 100 miles. 3 hours would be 150 miles, etc. To graph these you could use a table Time (h)DistanceOrdered pair 150(1,50) 2100(2,100) 3150(3,150) 4200(4,200) Let’s be y= the distance And x=the time(hours) The graphs of the ordered pairs(time,distance) lie on a line. The relationship between time and distance is linear. When data are linear; the rate is constant.

3. 0 1 2 3 4 5 6 300 50 100 150 200 250 Notice that the rate is just ratio of the rise over the run between two points. This rate of change is called slope

4. There is a worksheet for finding slope from the the graph only. KUTA and you have to do it now

5. Hours Worked (x) 1234 Dollars Earned (y) 10203040 If John earns $10.00 an hour, the above table shows the relationship between hours worked and dollars earned. From this table, we can see that dollars earned is equal to hours worked multiplied by 10. The equation y=10x describes this relationship. In this relationship, the number of dollars earned varies directly with the number of hours worked. Problem 2: Direct Variation

6. direct variation A direct variation is a relationship that can be represented by a function in the Form y=kx, where k‡0. the constant of variation for a direct variation k is the coefficient of x. By dividing each side by x you can see the ratio is constant

7. The money collected and the hours worked The distance traveled and the speed of the car The circumference of a circle and its diameter The area of a circle and its radius M=kh d=kt C=kd Examples of direct variation

8. Problem 1 : Does the equation represent a direct variation? Identifying a direct variation a) b) First:Solve the equation for y The equation has the Form Y=kx, so the equation is a direct variation. Its constant of variation is You cannot write the equation in the form y=kx. It is not a direct variation

9. Your turn Does 4x+5y=0 represent a direct variation, if so Find the constant of variation. Answer:

10. Graphing a direct variation Weight on Mars y varies directly with weight on Earth x. The weights of the Science instruments onboard the Phoenix Mars Lander on Earth and mars are shown. Weight on Mars 50 lbs and the weight on Earth 130lbs a)What is an equation that relates weight, in pounds on Earth x and weight on Mars y?

11. What is the graph? x0.38xy 00.38(0)0 500.38(50)19 1000.38(100)38 1500.38(150)57 Ordered pairs (x,y) (0,0) (50,19) (100,38) (150,57) 50 100 150 60 40 20

12. Your turn Weight on the moon y varies directly with the weight on Earth x. A person who weights 100lb on Earth weights 16.6 lb on the moon. What is an equation that relates weight on Earth x and weight on the moon y? what is the graph of this equation. 2 4 6 0.4 0.2 Answer:

13. Take a note: the graph of a direct variation equation y=kx is a line with the following properties. The line passes through (0,0) The slope of the line is k When k›0 when k‹0 x x y y

14. Writing a Direct Variation From a Table For the data in the table, does y vary directly with x? if it does, write an equation for the direct variation. xy 46 812 1015 xy -23.2 12.4 41.6 a) b) Findfor each ordered pair Y=1.5x So y does not vary directly

15. Your turn For the data in the table, does y vary directly with x? if it does, write an equation for the direct variation. xy -32.25 1-0.75 4-3 xy 35.4 712.6 1221.6 xy -69 1-1.5 8-12 A B C Answers