1. Graphs of Functions The graph of a function gives you a visual representation of its rule. A set of points generated like we did in the previous section. You can determine if a given graph is a true function by the VERTICAL LINE TEST. Given the graph of a relation, the vertical line test is used to check if that relation is a function. The vertical line CAN NOT touch the graph in more than 1 spot.

2. Graphs of Functions VERTICAL LINE TEST – a vertical line drawn anywhere on the graph CAN NOT intersect the graph more than once. Here is the graph of a relation. Is it a true function ?

3. Graphs of Functions VERTICAL LINE TEST – a vertical line drawn anywhere on the graph CAN NOT intersect the graph more than once. Here is the graph of a relation. Is it a true function ? YES…I can not draw a vertical line that passes thru the graph more than once

4. Graphs of Functions VERTICAL LINE TEST – a vertical line drawn anywhere on the graph CAN NOT intersect the graph more than once. Here is the graph of a relation. Is it a true function ?

5. Graphs of Functions VERTICAL LINE TEST – a vertical line drawn anywhere on the graph CAN NOT intersect the graph more than once. Here is the graph of a relation. Is it a true function ? NO…I only need to find one spot where a vertical line crosses more than one time.

6. Graphs of Functions One-to-One Functions – a function where each range value has one and ONLY one domain value. Horizontal line test checks for one-to-one… Here is the graph of a function. Is it one-to-one ?

7. Graphs of Functions One-to-One Functions – a function where each range value has one and ONLY one domain value. Horizontal line test checks for one-to-one… Here is the graph of a function. Is it one-to-one ? NO…I only need to find one spot where a horizontal line crosses more than one time.

8. Graphs of Functions One-to-One Functions – a function where each range value has one and ONLY one domain value. Horizontal line test checks for one-to-one… Here is the graph of a function. Is it one-to-one ?

9. Graphs of Functions One-to-One Functions – a function where each range value has one and ONLY one domain value. Horizontal line test checks for one-to-one… Here is the graph of a function. Is it one-to-one ? YES…I can not draw a vertical line that crosses more than one time.

10. Graphs of Functions One-to-One Functions – a function where each range value has one and ONLY one domain value. Horizontal line test checks for one-to-one… Functions that are not one-to-one have NO INVERSE !!!

11. Graphs of Functions Graphing functions falls into two categories… 1. Continuous or constant functions ( no breaks ) 2. Piecewise functions…breaks in the graph occur We will first look at the Continuous Functions… Steps :1. Create a table of values for f(x) 2. Plot the points and sketch your graph

12. Graphs of Functions Steps :1. Create a table of values for ƒ(x) 2. Plot the points and sketch your graph EXAMPLE :Graph ƒ(x) = x 2 – 5

13. Graphs of Functions Steps :1. Create a table of values for ƒ(x) 2. Plot the points and sketch your graph EXAMPLE :Graph ƒ(x) = x 2 – 5 xf(x) - 2 f(-2) = (-2) 2 -5 f(-2) = 4 – 5 f(-2) = -1

14. Graphs of Functions Steps :1. Create a table of values for ƒ(x) 2. Plot the points and sketch your graph EXAMPLE :Graph ƒ(x) = x 2 – 5 xf(x) - 2 - 4 f(-1) = (-1) 2 -5 f(-1) = 1 – 5 f(-1) = - 4

15. Graphs of Functions Steps :1. Create a table of values for ƒ(x) 2. Plot the points and sketch your graph EXAMPLE :Graph ƒ(x) = x 2 – 5 xf(x) - 2 - 4 0 - 5 f(0) = (0) 2 -5 f(0) = 0 – 5 f(0) = - 5

16. Graphs of Functions Steps :1. Create a table of values for ƒ(x) 2. Plot the points and sketch your graph EXAMPLE :Graph ƒ(x) = x 2 – 5 xf(x) - 2 - 4 0 - 5 1 - 4 f(1) = (1) 2 -5 f(1) = 1 – 5 f(1) = - 4

17. Graphs of Functions Steps :1. Create a table of values for ƒ(x) 2. Plot the points and sketch your graph EXAMPLE :Graph ƒ(x) = x 2 – 5 xf(x) - 2 - 4 0 - 5 1 - 4 2 - 1 f(2) = (2) 2 -5 f(2) = 4 – 5 f(2) = -1

18. Graphs of Functions Steps :1. Create a table of values for ƒ(x) 2. Plot the points and sketch your graph EXAMPLE :Graph ƒ(x) = x 2 – 5 xf(x) - 2 - 4 0 - 5 1 - 4 2 - 1 Plot the points…

19. Graphs of Functions Steps :1. Create a table of values for ƒ(x) 2. Plot the points and sketch your graph EXAMPLE :Graph ƒ(x) = x 2 – 5 xf(x) - 2 - 4 0 - 5 1 - 4 2 - 1 Sketch your graph…

20. Graphs of Functions Steps :1. Create a table of values for ƒ(x) 2. Plot the points and sketch your graph EXAMPLE :Graph ƒ(x) = x 3 + 2 xf (x) -3- 25 f(-3) = (-3) 3 + 2 f(-3) = - 27 + 2 f(-3) = -25

21. Graphs of Functions Steps :1. Create a table of values for ƒ(x) 2. Plot the points and sketch your graph EXAMPLE :Graph ƒ(x) = x 3 + 2 xf (x) -3- 25 -2-6 f(-2) = (-2) 3 + 2 f(-2) = -8 + 2 f(-2) = -6

22. Graphs of Functions Steps :1. Create a table of values for ƒ(x) 2. Plot the points and sketch your graph EXAMPLE :Graph ƒ(x) = x 3 + 2 xf (x) -3- 25 -2-6 1 f(-1) = (-1) 3 + 2 f(-1) = -1+ 2 f(-1) = 1

23. Graphs of Functions Steps :1. Create a table of values for ƒ(x) 2. Plot the points and sketch your graph EXAMPLE :Graph ƒ(x) = x 3 + 2 xf (x) -3- 25 -2-6 1 02 f(0) = (0) 3 + 2 f(0) = 0+ 2 f(0) = 2

24. Graphs of Functions Steps :1. Create a table of values for ƒ(x) 2. Plot the points and sketch your graph EXAMPLE :Graph ƒ(x) = x 3 + 2 xf (x) -3- 25 -2-6 1 02 13 f(1) = (1) 3 + 2 f(1) = 1 + 2 f(1) = 3

25. Graphs of Functions Steps :1. Create a table of values for ƒ(x) 2. Plot the points and sketch your graph EXAMPLE :Graph ƒ(x) = x 3 + 2 xf (x) -3- 25 -2-6 1 02 13 210 f(2) = (2) 3 + 2 f(2) = 8 + 2 f(2) = 10

26. Graphs of Functions Steps :1. Create a table of values for ƒ(x) 2. Plot the points and sketch your graph EXAMPLE :Graph ƒ(x) = x 3 + 2 xf (x) -3- 25 -2-6 1 02 13 210 329 f(3) = (3) 3 + 2 f(3) = 27 + 2 f(3) = 29

27. Graphs of Functions Steps :1. Create a table of values for ƒ(x) 2. Plot the points and sketch your graph EXAMPLE :Graph ƒ(x) = x 3 + 2 xf (x) -3- 25 -2-6 1 02 13 210 329 Plot your points…

28. Graphs of Functions Steps :1. Create a table of values for ƒ(x) 2. Plot the points and sketch your graph EXAMPLE :Graph ƒ(x) = x 3 + 2 xf (x) -3- 25 -2-6 1 02 13 210 329 Sketch the graph…