A graph is a diagram of a relationship of (at least) two variables with changing values.

3. A graph is a diagram of a relationship of (at least) two variables with changing values.

4. The current value of each variable is represented as a distance from the origin.

5. The coordination of the variables is represented as a point on the graph

6. All points on the graph are equally important.

7. A graph can just be a set of discrete points

8. {(0,1), (1.5,3), (2.3,π), (5,0.7), (5+π,6), (9,2)}

9. Or a shape…

10. The equation of a graph • The algebraic relationship between variables on a graph • The equation is test: If you pick a pair of values that makes the equation true, that point is on the graph. If you pick a pair that is not true, the point is not on the graph. • Graphing is finding all the points that make the equation true, and changing their color.

11. How would we find the equation of this graph?

12. Tool: The distance between two points

13. Tool: The distance between two points

14. Horizontal distance between points is x-a

15. Vertical distance between points is y-b

16. We have a right triangle.

17. Pythagorean Theorem

18. Distance Formula • The distance between (a,b) and (x,y) is

19. Back to a circle problem

20. Circle • A circle is the set of all points a given distance (the radius) from a given point (the center).

21. Our circle is all the points 4 units away from (1,2)

22. All points 4 units away from (1,2) For any point (x,y) The distance between (1,2) and (x,y) is 4 Using the Pythagorean Theorem… Is the equation of the circle

23. Equation of a circle • For radius r and center (a,b)

24. Consider a circle with equation x2 + (y+4)2 = 289. The center and radius are given by: • (0,4), radius=289 • (0,-4), radius=289 • (4,0), radius=289 • (-4,0), radius=289 • None of the above.

25. Consider a circle with equation x2 + (y+4)2 = 289. The center and radius are given by: E

26. Midpoint formula • The point halfway between (a,b) and (x,y) is

27. Functions

28. A function is a relationship between two changing variables • An “input” variable • An “output” variable • The result of “doing” the function to the output variable • Both variables change so that the “input” variable always tells you exactly what the “output” variable is. • You never get two outputs for the same input.

29. Function Output Any time I know the input, that’s enough information to tell me the output. Example: when input is 2, output is 20. input

30. Not a Function Input Just knowing the input is not enough to tell me the output Example: when input is 20, the output could be 2, 3.6, or 6.9. Output

31. Function output input

32. Not a Function output input

33. Function notation • For a function named ƒ • And an input variable named x • The output variable is named ƒ(x). • ƒ(x) is the number that is the result of doing the action ƒ to the number x

34. WARNING • ƒ(x) DOES NOT MEAN ƒ*x • You can only multiply numbers. f is NOT a number. f is the name of a relationship. • x and ƒ(x) are the numbers. • Brangelina is not a person. • Brangelina is the name of a relationship • Brad and Angelina are the people

35. How to do a function to the input number • Algebra: Substitute

36. How to do a function to the input number ƒ(x) ƒ x Find your input value on the x axis. Here the input value is 5.5

37. How to do a function to the input number ƒ(x) ƒ x Go up and over to Find your output value on the ƒ(x) axis. Here the output value is 25

38. How to do a function to the input number ƒ(x) ƒ x Input 5.5, output 25, name of function ƒ. ƒ(5.5)=25.

39. Given the function f(x)=x2 -2Mx, where M is some parameter, find f(3). • 3 • 3M • 9-6M • 6-9M • None of the above.

40. Given the function f(x)=x2 -2Mx, where M is some parameter, find f(3). C

41. Domain and Range • Function is a relationship between a changing input variable and a changing output variable. • The domain is a description of all the values that the changing input variable takes on. • The range is a description of all the value that the changing output variable takes on.

42. Example ƒ(x) Domain: [0,9] Range: [0,30] ƒ x

43. State the range of the function whose graph is pictured here. Select the best answer! A) B) C) D) E) None of the above

44. State the range of the function whose graph is pictured here. Select the best answer! D First, the output changes from -1 to -3 (excluding -1). The output can be any value between -3 and -1, but can’t be -1. Range: [-3,-1) Later, the output changes from 2 to 3. The output can be any value between 2 and 3, including 2 and 3. Additional Range: [2,3]