December 10, 2010

1. December 10, 2010 Functions and Relations

2. Warm Up

3. Frayer Model Examples Definition Relation Non-Linear Linear

4. Frayer Model Examples Definition A set containing pairs of numbers Relation Non-Linear Linear

5. Frayer Model Examples Definition {(2,1), (1,3), (0,4)} A set containing pairs of numbers 2 1 0 1 3 4 x y 2 1 0 1 3 4 Relation Non-Linear Linear

6. Frayer Model Examples Definition {(2,1), (1,3), (0,4)} A set containing pairs of numbers 2 1 0 1 3 4 x y 2 1 0 1 3 4 Relation Non-Linear Linear

7. Frayer Model Examples Definition {(2,1), (1,3), (0,4)} A set containing pairs of numbers 2 1 0 1 3 4 x y 2 1 0 1 3 4 Relation Non-Linear Linear

8. Frayer Model Examples Definition a relation in which each input (x value) is paired with exactly one output (y value). Function Non-Linear Linear

9. Frayer Model Examples Definition {(1,2), (2,4), (3,6)} a relation in which each input (x value) is paired with exactly one output (y value). 1 2 3 2 4 6 x y 1 2 3 2 4 6 Function Non-Linear Linear

10. Frayer Model Examples Definition {(1,2), (2,4), (3,6)} a relation in which each input (x value) is paired with exactly one output (y value). 1 2 3 2 4 6 x y 1 2 3 2 4 6 Function Non-Linear Linear

11. Frayer Model Examples Definition {(1,2), (2,4), (3,6)} a relation in which each input (x value) is paired with exactly one output (y value). 1 2 3 2 4 6 x y 1 2 3 2 4 6 Function Non-Linear Linear

12. Answer Now Is this relation a function?{(1,3), (2,3), (3,3)} • Yes • No

13. Are these relations functions? • (1, 2), (3, 4), (1, 5), (2, 6) • (6, 9), (7, 10), (8, 11), (8, –11) • (–1, –5), (–2, –7), (0, 3), (1, –5) 4. (2, 4), (3, 5), (2, -4), (3, –5)

14. Are these relations functions? 1. 2. 3.

15. Are these relations functions? 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

16. Vertical Line Test (pencil test) If any vertical line passes through more than one point of the graph, then that relation is not a function. Are these functions? FUNCTION! FUNCTION! NOPE!

17. Vertical Line Test FUNCTION! NO! NO WAY! FUNCTION!

18. Answer Now Is this a graph of a function? • Yes • No

19. An Equation is not a Function if… • the “y” variable is raised to an EVEN power; • x = any number; x = 5 and x = -9 – these are vertical lines.

20. y = x 2 x + y 4 = 5 y = x + 3 y + 6 = x 3 x = 3 y 3 = x 2 + 4 Yes No, y is raised to an even power! Yes Yes No, because the line is vertical! Yes Are these functions?

21. An Equation is NOT linear if… • any two variables are being multiplied together, Or… • there is a power on any variable greater than 1, Or… • there is a variable in the denominator.

22. 2a + 3b = 4 Are any two variables being multiplied together? Is there a power on any variable greater than 1? Is there a variable in the denominator? Since the answer to all of the questions is “no,” the equation is linear. No! No! No! Is the following equation linear?

23. y = 5x – 3xy Are any two variables being multiplied together? Is there a power on any variable greater than 1? Is there a variable in the denominator? Since the answer to the first question is “yes,” the equation is nonlinear. Yes! No! No! Is the following equation linear?

24. y = Are any two variables being multiplied together? Is there a power on any variable greater than 1? Is there a variable in the denominator? Since the answer to the third questions is “yes,” the equation is nonlinear. No! No! Yes! Is the following equation linear?

25. A = s 2 Are any two variables being multiplied together? Is there a power on any variable greater than 1? Is there a variable in the denominator? Since the answer to the second questions is “yes,” the equation is nonlinear. No! Yes! No! Is the following equation linear?

26. Is the table linear? 3 3 3 Yes! The “x” values increase by 1 and the “y” differences are the same. Therefore, this relationship is linear.

27. Is the table linear? 9 11 13 No! Though the “x” values increase by 1, the “y” differences are not the same. Therefore, this relationship is nonlinear.