Patterns and Proportional Relationships

1. Patterns and Proportional Relationships Standard: M7A3 Describe patterns in the graphs of proportional relationships, both direct (y = kx) and inverse (y = k/x).

2. Direct Proportion • The table above is a linear function. One way to describe the function is, as x increases by one, y increases by 8. Another way to describe the function is by looking at the ratio y/x in each column of the table. In each column y/x = 8. This is the same as rise/run, which equals slope. The equation is: y = 8x Any function (linear table) that can be written as y = kx, where k is a constant, is called a direct proportion.

3. Direct Proportion in a Graph • How can I determine if a graph is a direct proportion? • 1. The graph of a direct proportion must be a straight line and pass through the point (0, 0). • 2. Pick points on the graph and divide y by x (find slope). If the answer is a constant (k) each time, the graph fits the criteria for a direct proportion.

4. You try it! • Scout earns $12 an hour for each hour she works. Show that a direct proportion exists between the number of hours she works and the amount of money she earns. • Create a table using the information you have. • Write an equation representing the situation. • Does the equation for this problem fit the equation for a direct proportion y = kx ?

5. Indirect Proportion • An indirect proportion exists when the product (answer you get when you multiply) of x and y is a constant (k). • For example, an indirect proportion can be written as xy = k or as y = k/x. Why y = k/x? • The graph of an indirect proportion will never be a straight line and it will never go through (0,0).

6. Graph the following coordinate pairs • (1, 8) • (2, 4) • (4, 2) • (8, 1) • Is this a direct proportion or indirect proportion? • This graph is typical of an indirect proportion. • What happens when you multiply x and y? • Each product equal 8, thus this is an indirect proportion.

7. You try it! • Is the function y = 4x + 2 a direct proportion? • Why? Or Why not?

8. Is the following table of a direct or indirect proportion? indirect

9. Is the following table of a direct or indirect proportion? direct

10. Direct or indirect? y = 7x y = 7/x xy = 12 y/x = 5 direct indirect indirect direct

11. Is the graph direct or indirect? indirect

12. Is the graph direct or indirect? direct

13. Is the graph direct or indirect? neither

14. Is the graph direct or indirect? neither

15. Is the graph direct or indirect? neither

16. Is the graph direct or indirect? indirect

17. What happens to the quantities on the right as those on the left increase? • Speed of car time spent traveling • Number of workers time needed to finish a job • Length of a hammer effort needed to pull a nail • # of car payments amount due on car loan • These are all indirect proportions because as one quantity increases, the other decreases.

18. Jackie’s parents are being generous and decide to give her $120 to split evenly among herself and her friends to go to the amusement park. There is an indirect proportion between the number of friends going to the park and _____________. • $120 • The number of friends not going to the park • The amount of money each friend receives • The amount left over after the money is given out.

19. Jackie’s parents are being generous and decide to give her $120 to split evenly among herself and her friends to go to the amusement park. • Create a table and graph to show how much money each person would get if you split the money evenly.