1. Section 2.1 - Functions and Their Graphs
ALGEBRA TWO
Section Functions and Their Graphs

2. LEARNING GOALS Goal One - Represent relations and functions.
Goal Two - Graph and evaluate linear functions.

3. VOCABULARY
A relation is a mapping, or pairing, of input values with output values.
The set of input values is the domain, and the set of output values is the range.
A relation is a function provided there is exactly one output for each input. 1

4. Identifying Functions
Decide whether the relation shown in the input-output diagram is a function. If it is a function, give the domain and the range.
Input Output
The relation is NOT a function because the input 3 has two outputs: 8 and 10.

5. Identifying Functions
Decide whether the relation shown in the input-output diagram is a function. If it is a function, give the domain and the range.
The relation IS a function. For each input there is exactly one output. The domain of the function is the set of input values 1, 2, 3, and 4. The range is the set of output values 5 and 7.
b. Input Output 2 4

6. Identifying Functions
Decide whether the relation shown in the input-output diagram is a function. If it is a function, give the domain and the range.
Input Output 5
The relation is NOT a function. The input 4 has two outputs 3 and 5.

7. Identifying Functions
Decide whether the relation shown in the input-output diagram is a function. If it is a function, give the domain and the range.
The relation IS a function. For each input there is exactly one output. The domain of the function is the set of input values 1, 2, 3, and 4. The range is the set of output values 1, 4, 9, and 16.
Input Output

8. Identifying Functions
Decide whether the relation shown in the input-output diagram is a function. If it is a function, give the domain and the range.
The relation IS a function. For each input there is exactly one output. The domain of the function is the set of input values 1, 2, 3, and 4. The range is the set of output values 1, 4, 9, and 16.
Input Output 1

9. VOCABULARY
A relation can be represented by a set of ordered pairs of the form (x, y).
In an ordered pair, the first number is the x-coordinate and the second number is the y-coordinate.
To graph a relation, plot each of its ordered pairs in a coordinate plane. 1

10. VOCABULARY
A coordinate plane is divided into four quadrants by the x-axis and the y- axis.
The axes intersect at a point called the origin. 1

11. Construct a table of input and output values for the function.
GRAPHING A FUNCTION
You should recall from Algebra One that the simplest way to graph a function is to:
Construct a table of input and output values for the function.
Substitute various numbers for x and solve for y.
Plot the points on an x-y coordinate plane.
Draw the line connecting the points. 1

12. STEP ONE: CONSTRUCT A TABLE OF INPUT AND OUT VALUES FOR THE FUNCTION.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = (-1/2)x + 1
SOLUTION
STEP ONE: CONSTRUCT A TABLE OF INPUT AND OUT VALUES FOR THE FUNCTION. x y 1

13. STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = (-1/2)x + 1
SOLUTION
STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y. x -4 -2 2 4 y 1

14. STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = (-1/2)x + 1
SOLUTION
STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y. x -4 -2 2 4 y 3 1

15. STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = (-1/2)x + 1
SOLUTION
STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y. x -4 -2 2 4 y 3 1

16. STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = (-1/2)x + 1
SOLUTION
STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y. x -4 -2 2 4 y 3 1 1

17. STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = (-1/2)x + 1
SOLUTION
STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y. x -4 -2 2 4 y 3 1 1

18. STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = (-1/2)x + 1
SOLUTION
STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y. x -4 -2 2 4 y 3 1 -1 1

19. STEP THREE: PLOT THE POINTS.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = (-1/2)x + 1
SOLUTION
STEP THREE: PLOT THE POINTS.
(-4, 3), (-2, 2), (0, 1), (2, 0), & (4, -1) 1

20. STEP FOUR: DRAW A LINE THROUGH THE POINTS.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = (-1/2)x + 1
SOLUTION
STEP FOUR: DRAW A LINE THROUGH THE POINTS. 1

21. STEP ONE: CONSTRUCT A TABLE OF INPUT AND OUT VALUES FOR THE FUNCTION.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = x - 4
SOLUTION
STEP ONE: CONSTRUCT A TABLE OF INPUT AND OUT VALUES FOR THE FUNCTION. x y 1

22. STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = x - 4
SOLUTION
STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y. x 1 2 3 4 y 1

23. STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = x - 4
SOLUTION
STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y. x 1 2 3 4 y -4 1

24. STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = x - 4
SOLUTION
STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y. x 1 2 3 4 y -4 -3 1

25. STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = x - 4
SOLUTION
STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y. x 1 2 3 4 y -4 -3 -2 1

26. STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = x - 4
SOLUTION
STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y. x 1 2 3 4 y -4 -3 -2 -1 1

27. STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = x - 4
SOLUTION
STEP TWO: SUBSTITUTE VARIOUS NUMBERS FOR x AND SOLVE FOR y. x 1 2 3 4 y -4 -3 -2 -1 1

28. STEP THREE: PLOT THE POINTS.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = x - 4
SOLUTION
STEP THREE: PLOT THE POINTS.
(0, -4), (1, -3), (2, -2), (3, -1), & (4, 0) 1

29. STEP FOUR: DRAW A LINE THROUGH THE POINTS.
GRAPHING A FUNCTION
PROBLEM: Graph the function y = x - 4
SOLUTION
STEP FOUR: DRAW A LINE THROUGH THE POINTS. 1

30. EVALUATING FUNCTIONS
A function is written exactly like the equation for a line (y = mx + b) except that the symbol f (x) is substituted for y. Evaluating a function in the form f (x) = mx + b follows the exact same process as a linear equation. 1

31. VERTICAL LINE TEST
A relation IS a function if and only if no vertical line intersects the graph of the relation at more than one point. 1

32. VERTICAL LINE TEST
A relation IS a function if and only if no vertical line intersects the graph of the relation at more than one point. 1

33. VERTICAL LINE TEST
A relation IS a function if and only if no vertical line intersects the graph of the relation at more than one point. 1

34. VERTICAL LINE TEST
A relation IS a function if and only if no vertical line intersects the graph of the relation at more than one point. 1

35. VERTICAL LINE TEST
A relation IS a function if and only if no vertical line intersects the graph of the relation at more than one point.
This relation is a function because it passes the vertical line test. 1

36. VERTICAL LINE TEST
A relation IS a function if and only if no vertical line intersects the graph of the relation at more than one point. 1

37. VERTICAL LINE TEST
A relation IS a function if and only if no vertical line intersects the graph of the relation at more than one point. 1

38. VERTICAL LINE TEST
A relation IS a function if and only if no vertical line intersects the graph of the relation at more than one point. 1

39. VERTICAL LINE TEST
A relation IS a function if and only if no vertical line intersects the graph of the relation at more than one point.
Here the vertical line passes through the graph in three places!!!! 1

40. VERTICAL LINE TEST
A relation IS a function if and only if no vertical line intersects the graph of the relation at more than one point.
This relation is NOT a function because it fails the vertical line test. 1

41. VERTICAL LINE TEST
A relation IS a function if and only if no vertical line intersects the graph of the relation at more than one point.
Here the vertical line passes through the graph in three places!!!! 1

42. Review
In this section we have seen three ways to tell if a relation is a function
1. If given inputs and outputs: check that each input has only one output
2. If given a graph: use the vertical line test (no vertical intersects the graph of the relation at more than one point)
3. If given an equation: so far we know all equations that can be written in the form y=mx+b are functions, and any equations were x2 is present, is not (there are more, but for now these are obvious signs to look for)

43. ASSIGNMENT READ & STUDY: pg. 67-70.
WRITE: pg #19, #23, #25, #27, #31, #35, #41, #43, #47, & #51.