1. 1.3 Families of Equations

2. What families of graphs have your studied? Linear Absolute Value Quadratic Square Root Cubic Cube Root

3. How do you recognize the various families? Linear Family Equation: Graph: y = mx + b Ax + By + C = 0 Line: m = + m = – m = 0

4. Examples: Graph the following linear equations: y = 3x - 5 Domain-- Range-- Increasing/Decreasing? Odd/Even

5. Examples: Graph the following linear equations: 2x – 3y = 12 Domain-- Range-- Increasing/Decreasing? Odd or Even?

6. How do you recognize the various families? Absolute Value Family Equation: Graph: y = a|x – h| + k (h,k) a = – a = +

7. Examples: Graph the following absolute value equations: y = |x| Domain-- Range-- Increasing/Decreasing? Odd or Even?

8. Examples: Graph the following absolute value equations: y = |x - 2|+ 3 Domain-- Range-- Increasing/Decreasing? Odd or Even

9. Examples: Graph the following absolute value equations: y = |x + 6|+ 5 Domain-- Range-- Increasing/Decreasing? Odd or Even

10. Examples: Graph the following absolute value equations: y = -|x - 1|-3 Domain-- Range-- Increasing/Decreasing? Odd or Even?

11. How do you recognize the various families? Quadratic Family Equation: Graph: Pattern: y = a(x – h) 2 + k y = ax 2 + bx + c a = – (h,k) a = + (h,k) 1, 3, 5

12. Examples: Graph the following quadratic equations: y = x 2 Domain-- Range-- Increasing/Decreasing? Odd or Even

13. Examples: Graph the following quadratic equations: y = (x-1) 2 Domain-- Range-- Increasing/Decreasing? Odd or Even?

14. Examples: Graph the following quadratic equations: y = -(x-2) 2 + 4 Domain-- Range-- Increasing/Decreasing Odd or Even?

15. Examples: Graph the following quadratic equations: y = x 2 + 4x - 1 Hint: x = -b 2a

16. How do you recognize the various families? Square Root Family Equation: Graph: Pattern: a = – a = + y = a√x – h + k (h,k) or (h,k) or 1, 3, 5 y = a(x – h) 1/2 + k

17. Examples: Graph the following square root equations: y = √x Domain-- Range-- Increasing/Decreasing

18. Examples: y = √x – 3 + 2 Graph the following square root equations: Domain-- Range-- Increasing/Decreasing

19. How do you recognize the various families? Cubic Family Equation: Graph: Pattern: y = a(x – h) 3 + k a = – (h,k) a = + (h,k) 1, 7

20. Examples: Graph the following cubic equations: y = x 3 Domain-- Range-- Increasing/Decreasing

21. Examples: Graph the following cubic equations: y = x 3 - 1 Domain-- Range-- Increasing/Decreasing?

22. Examples: Graph the following cubic equations: y = -(x-2) 3 + 4 Domain-- Range-- Increasing/Decreasing?

23. How do you recognize the various families? Cube Root Family Equation: Graph: Pattern: a = – a = + y = a 3 √x – h + k (h,k) 1, 7 y = a(x – h) 1/3 + k

24. Examples: y = 3 √x Graph the following cube root equations: Domain-- Range-- Increasing/Decreasing

25. Examples: y = 3 √x – 3 + 2 Graph the following cube root equations: Domain-- Range-- Increasing/Decreasing?

26. Identify the family of graphs for the following equations: EquationFamily 3x + 2y = 6 y = (x – 2) 2 + 1 y = | x – 4 | + 6 y = 3x 2 + 6x – 2 y = 2  x – 1 + 4 y = – 3  4 – x + 2 (x – 2) 2 + (y – 6) 2 = 4

27. Identify the family of graphs for the following equations: EquationFamily y = 2 / 3 | x – 4 | + 1 y =  1 – x x = 5 x 2 +y 2 + 3x -1 = 0 x 2 + y 2 = 84 y = – 3  x + 1 – 3 y = – 2

28. Identify the family of graphs for the following equations: EquationFamily y = – ( x – 4 ) 2 y = 2x 2 + 1 y = 3 / 2 x – 2 y = –2 | x | y = (x – 8) 1/3 - 3

29.  Average Rate of Change f(x 2 ) – f(x 1 ) x 2 - x 1 Change in y Change in x m sec

30.  A function y = f(x) is even if f(-x) = f(x)  A function y = f(x) is odd if f(-x) = -f(x)