2. Graphing Lines

3. How To Find THE SLOPE of a Line Given Two Points How To Find THE SLOPE of a Line Given Two Points

4. Slope of a Line x y x1x1 x2x2 y2y2 y1y1

6. Definition of Slope The slope of the line is it’s measure of steepness. It measures the rate of change of the line. In all lines the slope is constant, it doesn’t change no matter where you are at on the line.

7. Find the Slope of the line containing the points

8. Bicycles In groups, generate the line that relates the number of wheels to the number of bicycles. Represent it Graphically. What does the slope of this line represent?

9. Graph of a Line xy -5 0-3 1 21 33

10. Intercepts Where the graph crosses the x-axis is the x-intercept. It has coordinates (a,0). Where the graph crosses the y-axis is the y-intercept. It has coordinates (0,b).

11. y x x-intercept y-intercept (a,0) (0,b) The x and y intercepts of a line

12. Example Find the x and y intercepts of the line given by:

13. Slope Intercept Form x y Any stinking point on the line y-intercept

14. Determine the slope and y-intercept of the following

15. Point Slope Form x y Any stinking point on the line Any stinking fixed point on the line

16. Find the Equation of the Line Through the Points

17. Line Formulas Slope Formula Slope Intercept Form Point Slope Form

18. Definition of Function or How to Relate

19. Definition of a Relation

20. Relation (1)32 mpg (2)8 mpg (3)16 mpg (A) (C) (B)

21. Domain and Range The values that make up the set of independent values are the domain The values that make up the set of dependent values are the range. State the domain and range from the 4 examples of relations given.

22. DomainRange Correspondence or Relation

23. Definition of a Relation A Relation maps a value from the domain to the range. A Relation is a set of ordered pairs. The most common types of relations in algebra map subsets of real numbers to other subsets of real numbers.

24. Example DomainRange 3π 11- 2 1.6182.718

25. Definition of a Function If a relation has the additional characteristic that each element of the domain is mapped to one and only one element of the range then we call the relation a Function.

26. Definition of a Function If we think of the domain as the set of boys and the range the set of girls, then a function is a monogamous relationship from the domain to the range. Each boy gets to go out with one and only one girl. But… It does not say anything about the girls. They get to live in Utah.

27. Decide if the Relation is a Function. The relation is the year and the cost of a first class stamp. The relation is the weight of an animal and the beats per minute of it’s heart. The relation is the time of the day and the intensity of the sun light. The relation is a number and it’s square. The relation is time since you left your house for work and your distance from home.

28. Examples Please Give three examples from the real world of relations. Be sure and state the domain, the range, and the definition of how the variables are related. Decide which if any of your examples are functions.

29. x DOMAIN y1y2y1y2 RANGE R NOT A FUNCTION

30. y RANGE f FUNCTION x1x1 DOMAIN x2x2

31. Mathematical Examples Decide if the following relations are functions. X Y 1 2 -5 7 -1 2 3 X Y 1 -5 1 -1 1 3 1 X Y 1 2 1 7 1 2 1 3 X Y 1 π π 1 -1 5 π 3

32. Ways to Represent a Function SymbolicSymbolic X Y 1 2 5 10 -1 -2 3 6 GraphicalGraphical NumericNumeric VerbalVerbal The cost is twice the original amount.

33. Function Notation The Symbolic Form A truly excellent notation. It is concise and useful.

34. Output Value Member of the Range Dependent Variable These are all equivalent names for the y. Input Value Member of the Domain Independent Variable These are all equivalent names for the x. Name of the function

35. Examples of Function Notation The f notation Find f(2), g(-1), f(-0.983),

36. Your Turn! Given: Evaluate the following:

37. Graphical Representation Graphical representation of functions have the advantage of conveying lots of information in a compact form. There are many types and styles of graphs but in algebra we concentrate on graphs in the rectangular (Cartesian) coordinate system.

38. Graphs and Functions Domain Range

39. Determine the Domain and Range for Each Function From Their Graph

40. Vertical Line Test for Functions If a vertical line intersects a graph once and only once for each element of the domain, then the graph is a function.

41. How to determine Domain and Range of a function. Graph the following on your calculator. Also give the algebra.

42. Big Deal! A point is in the set of ordered pairs that make up the function if and only if the point is on the graph of the function.A point is in the set of ordered pairs that make up the function if and only if the point is on the graph of the function.

43. Key Points Definition of a function Ways to represent a function Symbolically Graphically Numerically Verbally