1. y = (1) – 2 y = 1 – 2 y = -1 y = x – 2 x y y = (–1) – 2 y = -3 –1 –3
Graph the linear equation.
y = x – 2
1. Draw a table & Choose 3 values for x and solve for ‘y’.
Evaluate for x= ( -1, 0, 1)
y = (1) – 2
y = 1 – 2
y = -1 x
y = x – 2 y
y = (–1) – 2
y = -3 –1 –3 –2 1 -1
y = (0) – 2
y = –2
2. Plot points.
3. Draw a line through all points to show all ordered pairs that satisfy the equation.
REMEMBER: Draw arrows on both “ends” of the line. 1

2. y= (1) + 2 y = 3 y = x + 2 x y y = (–1) + 2 y = 1 –1 1 2 1 3 y=(0) + 2
Graph the linear equation.
1. Draw a table & Choose 3 values for x and solve for ‘y’.
Evaluate for x= ( -1, 0, 1)
y= (1) + 2
y = 3 x
y = x + 2 y
y = (–1) + 2
y = 1 –1 1 2 1 3
y=(0) + 2
y = 2
2. Plot points.
3. Draw a line through all points to show all ordered pairs that satisfy the equation.
REMEMBER: Draw arrows on both “ends” of the line. 2

3. Name the ordered pairs for each letter.6 A
A = (-6, 6)
B = (-5,2)
C = (0,1)
D = (4,3)
E = (-6,-1)
F = (-5,-5)
G = (-1,-2)
H = (1,-5)
I = (4,-3) 5 4 3 D 2 B 1 C -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 E -1 G -2 -3 I -4 F H -5 -6 3

4. Domain &
range
mapping 1 5 2 7 11 -1

5. In this relation: -5 is mapped onto 12, 1 is mapped onto 18,
mapping - an illustration how each element of the domain is paired with an element in the range.
In this relation:
-5 is mapped onto 12,
1 is mapped onto 18,
100 is mapped onto 117. 5

6. Relation  {(1, 0), (5, 2), (7, 2), (-1, 11)} Domain  {1, 5, 7, -1}
mapping - an illustration how each element of the domain is paired with an element in the range. 1 5 2 7 11 -1
Relation 
{(1, 0), (5, 2), (7, 2), (-1, 11)}
Domain 
{1, 5, 7, -1}
Range 
{0, 2, 11} 6

7. DETERMINING
IF IT IS A
FUNCTION
Given MAPPINGS

8. (domain does not repeat, range)
Each domain value is paired with exactly 1 range value.
FUNCTIONS
NOT FUNCTIONS
Many to One
Relationship
One to One
One to Many
Relationship 8

9. Give the domain and range of each relation
Give the domain and range of each relation. Tell whether the relation is a function and explain.
a. {(8, 2), (–4, 1),
(–6, 2),(1, 9)} b.
D: {–6, –4, 1, 8}
R: {1, 2, 9}
D: {2, 3, 4}
R: {–5, –4, –3}
The relation is a function. Each domain value is paired with exactly one range value.
The relation is not a function. The domain value 2 is paired with both –5 and –4.

10. Identifying Functions
Give the domain and range of the relation. Tell whether the relation is a function. Explain. –4
Use the arrows to determine which domain values correspond to each range value. 2 –8 1 4 5
D: {–4, –8, 4, 5}
R: {2, 1}
This relation is a function. Each domain value is paired with exactly one range value.

11. Relations Domain = {-3, 1,3,4} Range = {3,1,-2}
A relation is a mapping, or pairing, of input values with output values.
The set of input values is called the domain.
The set of output values is called the range.
A relation as a function provided there is exactly one output for each input.
It is NOT a function if at least one input has more than one output
Identify the Domain & Range. Then tell if the relation is a function.
Input Output 4
Domain = {-3, 1,3,4}
Range = {3,1,-2}
Function?
Yes: each input is mapped onto exactly one output 11

12. Identify the Domain and Range. Then tell if the relation is a function.
Input Output 4
Domain = {-3, 1,4}
Range = {3,-2,1,4}
Notice the set notation!!!
Function?
No: input 1 is mapped onto
Both -2 & 1 12

13. Give the domain and range of the relation
Give the domain and range of the relation. Tell whether the relation is a function. Explain.
D: {5, 10, 15};
R: {2, 4, 6, 8};
The relation is not a function since 5 is paired with 2 and 4.

14. Tests for function Ordered pairs or mappings
1st value of ordered pairs should be different.
2. vertical line test
vertical line cannot intersect a graph at in more than one point

15. Is the relation a function?
{(2,4),(3,6),(4,8)}
{(2,4),(3,4),(4,4)}
{(2,4),(2,6),(2,8)}
{(2,5),(3,7),(4,9)}
YES
YES NO
YES 15

16. Write the domain and range of each relation.
{(-2,-1), (0,1), (2,3), (4,5)}
{-2,0,2,4}; {-1,1,3,5}
2. {(0.5,-4), (0.75, -2), (1.25,-1)}
{0.5,0.75, 1.25}; {-4, -2,-1}
Determine whether each relation is a function. 3. x 1 2 3 4 y
YES 4. x -1 -2 y 2 -4 NO 5. x 1 2 3 y 4
YES

17. 5-Minute Check
Write the domain and range of the relation. Then determine
whether each relation is a function. 1) x 4 1 3 6 y 2 2)
D= {4,1,3,6}; R= {2,3,4}; yes
{(4,2), (-3,2), (8,2), (8,9), (7,5)}
D={4,-3,8,7}; R= {2,9,5}; no

18. Determine whether each relation is a function. Why?
Each element of the domain only has one element of the range.
There are no repeats in the domain. No repeats in the domain means this is a function.
List relations in ordered pairs:
{(3, 0), (4, -6), (-3, 10), (-8, 10)}
no repeats in the domain.
arrows leaving domain don’t repeat.
No repeats in domain – it is a function. B. X Y 3 4 -3 -8 -6 10
5 in the x pairs up with both 2 and 3 in the y.
one domain pairs with two different ranges.
a repeat in the domain.
A repeat in domain - not a function. C.

19. DOMAIN RANGE
REAL
LIFE

20. relation - set of ordered pairs (like the ones from the example)
New terms:
domain - the set of first coordinates of the ordered pairs
This is also considered the independent variable.
It changes due to human input or time.
In math, it is the x-value.
range - the set of second coordinates of the ordered pairs
This is also considered the dependent variable.
It changes because the domain (independent variable) changed.
In math, it is the y-value.
Remember, the domain is the x-value - the independent variable.
The range is the y-value - the dependent variable.

21. 5-2: Relations
In the U.S., the manatee, an aquatic mammal, is considered endangered.
In the table below, the number of manatees that have been found dead since 1981.

22. The manatee data can be represented by a set of ordered pairs (below
The manatee data can be represented by a set of ordered pairs (below.} (year, # of manatees)
{(1981, 116), (1982, 114), (1983, 81), (1984, 128), (1985, 119),
(1986, 122), (1987, 114), (1988, 133), (1989, 168), (1990, 206),
(1991, 174), (1992, 163), (1993, 145), (1994, 193), (1995, 201),
(1996, 415), (1997, 242), (1998, 231)}
Each ordered pair can then be graphed.

23. Relations Identify the domain & range from the manatee data.
All of the dates (1st values)are in the domain.
{(1981, 116), (1982, 114),
(1983, 81), (1984, 128),
(1985, 119), (1986, 122),
(1987, 114), (1988, 133),
(1989, 168), (1990, 206),
(1991, 174), (1992, 163),
(1993, 145), (1994, 193),
(1995, 201), (1996, 415),
(1997, 242), (1998, 231)}
{(1981, 116), (1982, 114),
(1983, 81), (1984, 128),
(1985, 119), (1986, 122),
(1987, 114), (1988, 133),
(1989, 168), (1990, 206),
(1991, 174), (1992, 163),
(1993, 145), (1994, 193),
(1995, 201), (1996, 415),
(1997, 242), (1998, 231)}
{(1981, 116), (1982, 114),
(1983, 81), (1984, 128),
(1985, 119), (1986, 122),
(1987, 114), (1988, 133),
(1989, 168), (1990, 206),
(1991, 174), (1992, 163),
(1993, 145), (1994, 193),
(1995, 201), (1996, 415),
(1997, 242), (1998, 231)}
The # of manatee (2nd values) are in the range.
Domain & range can be listed:
D: {1981, 1982, 1983, 1984, 1985, 1986, 1987, 1988, 1989, 1990, 1991, 1992, 1993, 1994, 1995, 1996, 1997, 1998}
R: {116, 81, 128, 119, 122, 133, 168, 206, 174, 163, 145, 193, 201, 415, 242, 231}
There are many ways to represent data. We looked at the manatee data as ordered pairs, a table, and a graph.
Data can also be represented in a mapping.

24. Relations EX: Represent the relation shown in the graph below as:
A. a set of ordered pairs,
B. a table, and
C. a mapping.
D. The determine the domain & range.
Determine coordinates of each point.
Write the ordered pairs.
Start on the left and work your way right.
{(-3, 3),
(-1, 2),
(1, 1),
(1, 3),
(3, -2),
(4, -2)}
B. Make a table and
put the values in:
C. Write the x’s -3 -1 1 3 4
Write the y’s 3 2 1 -2
Draw arrows to match pairs.
D. D = {-3, -1, 1, 3, 4}
R = {-2, 1, 2, 3}
NOTICE: There are no repeat between the domain and range. They are
listed in order from least to greatest. 24

25. Express the relation {(1, 3), (2, 4), (3, 5)} as a table, as a graph, and as a mapping diagram.
Use the x- & y-values to plot the ordered pairs.
Mapping Diagram x y
x y
Graph 1 3 1 3 2 4 2 4 3 5 3 5
x-values under “x”
y-values under “y”.
Write all x-values under “x” & all y-values under “y”. Draw an arrow from each x-value to its corresponding y-value.

26. Give the domain and range of the relation.
domain value is all x-values from 1 through 5, inclusive.
Domain: 1 ≤ x ≤ 5
range value is all y-values from 3 through 4, inclusive.
Range: 3 ≤ y ≤ 4 1 2 6 5
The domain values are all x-values 6, 5, 2, and 1. –4 –1
Domain: {6, 5, 2, 1}
The range values are y-values –4, –1, and 0.
Range: {–4, –1, 0}

27. Determine whether each relation is a function. Why?
There are no repeats in the domain. - is a function.
{(3, 0), (4, -6), (-3, 10), (-8, 10)}
no repeats in the domain.
arrows leaving domain don’t repeat.
it is a function. B. X Y 3 4 -3 -8 -6 10
5 in the x pairs up with both 2 and 3 in the y.
one domain pairs with two different ranges.
a repeat in the domain.
A repeat in domain - not a function. C.

28. Express the relation {(2, 3), (4, 4), (6, 8)} as a table and as a mapping diagram and identify if it is a function. List the domain and range.
x y
Table
Mapping Diagram x y 2 4 6 3 7 8 2 6 4 3 8 4 4
is a function
Domain (2,4,6) Range (3,4,8,)

29. Express the relation {(1, 3), (2, 5), (3, 5)} as a table and as a mapping diagram. Identify the domain and range.
Table
Mapping Diagram
x y x y 1 3 1 3 2 5 2 5 3 5 3
is a function
Domain (1,2,3) Range (3,5,)

30. Relations
Express the relation shown in the mapping as a set of ordered pairs. Identify the domain and range. Then write the inverse of this relation. X Y -1 3 4 -6 -4 2
Relation = {(3, 2), (4, -6), (3, -4), (-1, -6)}
Domain = {-1, 3, 4}
Range = {-6, -4, 2}
Inverse = {(2, 3), (-6, 4), (-4, 3), (-6, -1)}
BIG HINT:
Your 3-Strikes and Homework should look like this answer!!! 30

31. 5.2.2 DEFINITION OF THE INVERSE OF A RELATION
5-2: Relations
Relation Q is the inverse of relation S if and only if for every ordered pair (a, b) in S, there is an ordered pair (b, a) in Q.
DEFINITION OF THE INVERSE OF A RELATION
EXAMPLE 2: Express the relation shown in the mapping as a set of ordered pairs. Write the inverse of the relation and draw a mapping to model the inverse. X 1 2 3 4 Y -3 -2 -1
The mapping shows the relation
{(1, -3), (2, -2), (3, -1), (4, -1)}.
To get the inverse, switch the x- and y-values:
{(-3, 1), (-2, 2), (-1, 3), (-1, 4)}.
Then map the inverse. -3 -2 -1 1 2 3 4 31

32. EX: Solve y = x + 6. If the domain is {-4, -3, -1, 2, 4}
EX: Solve y = x + 6. If the domain is {-4, -3, -1, 2, 4}. Graph the solution set.
This is
a function
Make table with given x-values.
Determine the y-values.
Write solutions as ordered pairs.
Then plot the points.
The solution set is:
{(-4, 2), (-3, 3), (-1, 5), (2, 8), (4, 10)}. 32

33. Warm Up 1. 5x + 0 = –10 Solve each equation. –2 2. 33 = 0 + 3y 11 3. 1
4. –5y – 1 = 7y + 5
5. 6(x – 1) where x = 5 24
(5 – 1)s where s = –4 –6
7. Evaluate when x=2, y=3, and n=4
8. Evaluate y = for x={–10, –5, 0, 5, 10} 33

34. WARM UP SOLUTION #5 Evaluate when x=2, y=3, and n=4C
Substitute in the values for the variables
Exponents (powers) 33 = (3)(3)(3) = 27
Add:
Subtract
Exponents (powers) 62 = (6)(6) = 36
Subtract
Add 34

35. Warm Up 6. Evaluate y = for x={–10, –5, 0, 5, 10} y= (–10) y= (–5) y=
=–1 y=
(–10) y=
(–5)
= 0 y=
(0)
= 1 y=
(5)
= 2 y=
(10)
= 3 35

36. LINEAR EQUATION
SOLUTION SETS 36

37. 2y – x = 7 Original Equation 2(3) – (–1) = 7 Substitute for x and y.
Solution of an equation w/ one variable (one # which makes the equation true.
Is 3 a solution of 2x + 3 = 11?
2(3) + 3 = 11
Substitute 3 for x.
6 + 3 = 11
False equation
3 is not a solution
of 2x + 3 = 11.
Is 4 a solution of 2x + 3 = 11?
Substitute 4 for x.
2(4) + 3 = 11
True equation
4 is a solution of
2x + 3 = 11.
8 + 3 = 11
Linear equation with two variables (x, y)
1. Substitute: x value in place of x
y value in place of y.
(x, y)
(–1, 3)
2y – x = 7 Original Equation
2(3) – (–1) = 7 Substitute for x and y.
7 = Equation is satisfied. 37

38. SOLUTIONS TO EQUATIONS WITH 2 VARIABLES
SUBSTITUTION & SOLVE to determine whether the equation is true
x + y = 3 substitute (0, 3)
(0) + (3) = 3
3 = TRUE STATEMENT
*If they are true, then the coordinates are an ordered pair of the linear equation. 38

39. Substitute the ordered pair in the linear equation y= ½ x – 2 and solve.
(7,2)
(1,1)
(8,2)
(-4,-6)
(-4,-4)
(-6,1)
(0,-6)
(-8,-6)
(8,-2)
(0,0)
(0,-2)
(4,4)
(6,1)
(2,-3)
(-6,-5)
(-1,-5)
(-2,-3)
(6,3)
(3,-2)
(1,-1)
(4,-1)
(5,0)
(4,0)
(-1,-3)
(-7,-4)
(0,-5)
(-2, 0) 39

40. Graph your ordered pair. If it was true, use blue. If false, use red.
y= ½ x – 2
Equations with two variables, have ordered pairs (x, y) as solutions.
Ordered pair solutions can be presented on a graph.
graphing the equation (graphing all solutions (ordered pairs) of an equation on a graph.) 40

41. Ordered pairs which are solutions to y= ½ x – 2 are blue.
(7,2)
(1,1)
(8,2)
(-4,-6)
(-4,-4)
(-6,1)
(0,-6)
(-8,-6)
(8,-2)
(0,0)
(0,-2)
(4,4)
(6,1)
(2,-3)
(-6,-5)
(-1,-5)
(-2,-3)
(6,3)
(3,-2)
(1,-1)
(4,-1)
(5,0)
(4,0)
(-1,-3)
(-7,-4)
(0,-5)
(-2, 0) 41

42. (0) = 2(0) (-8) = 4(-2) 0 = 0 -8 = -8 (4) = 4(1) +0 4=4
Solution sets for Linear Equations
ordered pairs that make an equation true will form a line
Try this with the other points on this graph.
(0 , 0)
(x , y)
(-2 , -8)
y = 4x
y = 2x
(0) = 2(0)
(-8) = 4(-2)
(1 , 4)
(x , y)
0 = 0
-8 = -8
y = 4x +0
(4) = 4(1) +0
4=4
Are there any points in-between these points on the graph?
YES -These are points of a linear equation.
“ordered pairs that make an equation true will form a line”
This means:
all points lying on same line are a solution to linear equation.
linear equation with 2 variables (equation whose graph is a line.) 42

43. Determining Whether a Point is on a Graph
Solution of an equation with two variables is the set of all points (x, y) whose coordinates satisfy the equation.
Determining Whether a Point is on a Graph
Tell whether is on the graph of 2x + 5y = 16.
(3, 2)
Substitute:
2x y = 16
2(3) + 5(2) 16 = ? = ?
(3, 2) is a solution to 2x + 5y = 16,
(3, 2) is on the graph of the line. 
16 = 16
(8, 0)
2x + 5y = 16
Substitute:
2x + 5y = 16
(2, 2)
2(8) + 5(0) 16 = ?
2(2) + 5(2) 16 = ? = ? = ? 
16 = 16
14  16 
Since (8, 0) is a solution
(8, 0) is on the graph.
(2, 2) is not a solution to 2x + 5y = 16,
(2, 2) is not on the graph. 43

44. Determining Whether a Point is on a Graph
Without graphing tell whether each point is on the graph: x – 3y = 12
(1.5, –3.5)
(5, 1)
(0, –4)
Substitute:
x – 3y = 12
5 – 3(1) = ?
(5, 1) is not a solution
& is not on the graph.
5 – = ?
2  12 
Substitute:
x – 3y = 12
0 – 3(–4) 12 = ?
(0, –4) is a solution to x – 3y = 12, (0, –4) is on the graph. = ? 
12 = 12

45. Determining Whether a Point is on a Graph
Without graphing tell whether each point is on the graph of x – 3y = 12.
(1.5, –3.5)
Substitute:
x – 3y = 12
1.5 – 3(–3.5) 12 = ? = ? 
12 = 12
Since (1.5, –3.5) is a solution to x – 3y = 12, (1.5, –3.5) is on the graph. 45

46. FINDING
SOLUTIONS TO
LINEAR EQUATION 46

47. (1 , 4) is a solution List ordered pair
Finding solution sets for Linear Equations
1. Choose any value to substitute for x
2. solve for y.
STEPS
y = 4x Original Equation
y = 4(1) Substitute for x
y = Solve for y.
If x = -2,
then
what does
y = ?
(1 , 4) is a solution List ordered pair
If x = 1,
then
what does
y = ?
y = 4x Original Equation
y = 4(-2) Substitute for x
y = Solve for y.
(-2 , -8) is a solution List ordered pair 47

48. Finding solutions to equations
EVALUATING EQUATIONS
Think of a linear equation as an input-output machine.
For y = 5x, ‘input’ a value for x, the ‘output’ is 5x. 6 EX
Input 6 for x and find the output.
y = 5x
y = 5(6)
y = 30 30 48

49. Finding solutions to equations
Evaluate the function for the given input values.
Find y when x = 7
Find y when x = -4
y = 3(x) + 2
y = 3(x) + 2
y = 3(–4) + 2
Substitute
–4 for x.
Substitute
7 for x.
y = 3(7) + 2
y= –12 + 2 y=
Simplify.
Simplify.
y= –10
y= 23
solution set?
(-4,-10)
(7, 23)
For y = 5x – 5
Find y when x = 6
Find y when x = -2
y = 5x – 5
y = 5x – 5
y = 5(6) – 5
y = 5(–2) – 5
Y = –10 – 5
y= 30 – 5
y= 25
y= –15
solution set?
(6, 25)
(-2,-15) 49

50. Evaluate the linear equation for the given values.
Find ‘y’ when x = –3.
Find ‘y’ when x = 1.
y = 2x – 1
y = 2x – 1
y = 2(1) – 1
y = 2(–3) – 1
y= 2 – 1
y= –6 – 1
y= –7
y= 1
solution set?
(1, 1)
(-3,-7) 50

51. Is there a relationship between your height and shoe size?45 42 39 36 33 30 27
Can you predict how
big a foot someone
would be with a
height of 96 inches?
Foot length (cm) 24 21 18 15 12 9 6 3 8 16 24 32 40 48 56 64 72 80 88 96
104
height (inches) 51

52. TRY THESE
Write four solutions for each equation. Write the solutions as ordered pairs.
y = 2x – 4
y = x/2 – 3
x + y = 12
Determine whether each relation is linear.
x – y = 32
25 – y2 = x2

53. 1.
Tell whether is a solution of ( –
8, 7 ) 3x + y = –
17.
Yes
List four solutions of the equation. 2. 2 y 5x = + – , ( )
0, 2
1, 7
2, 12
– 1, – 3 10 y 4x = + 3. ( ),
1, 6
0, 10
–1, 14 )
2, 2
List four solutions of the equation. 4. 2 y 6x = – ( ),
0, – 2
1, 4 )
2, 10
– 1, – 8
Which of the points lies on the line 2x + 3y = 6? A. B. C. D.
(0, 2)
(0, 3)
(1, 2)
(3, 1)

54. SOLUTION
SETS
DETERMINE
POINTS ON
A GRAPH

55. SYNONYMS
FUNCTIONS
GRAPHING
relation
Domain
range
Function
ordered pairs (x,y)
x-value
y-value
Linear equation

56. GRAPH of a Linear Equation
drawing that represents the solution set. (A LINE)
SOLUTION SET: (the list of solutions)
ORDERED PAIRS are solutions for Linear Equations. y
(0, 2)
(–2, 0) 2
(4 , 0)
y = x + 2 x –3 1 2 3
y = x substitute (0, 2) (2) = (0) + 2
2 = TRUE STATEMENT
y = x substitute (4, 0) (0) = (4) + 2
0 = FALSE!
(–2, 0) and
(0, 2) are part of the solution set for linear equation y = x + 2

57. SOLUTIONS TO EQUATIONS WITH 2 VARIABLES
SUBSTITUTION & SOLVE to determine whether the equation is true
y = 3 + x substitute ( 5 , 2)
y = 3 + x substitute (-3, 0)
y = 3 + x substitute (2, 5)
y = 3 + x substitute (3, 0)
*If they are true, then the coordinates are an ordered pair of the linear equation.
y = x substitute (0, 2)
(2) = (0) - 5
2 = FALSE!
FALSE
TRUE
TRUE
FALSE

58. Determining Whether a Point Is on a Graph
Without graphing, tell whether the point
is on the graph of
(–2, 10)
Substitute (–2, 10) into = ? = ? 
10 = 10
Since (–2, 10) is a solution of ,
(–2, 10) is on the graph.

59. Determining Whether a Point Is on a Graph
Without graphing, tell whether the point
is on the graph of
(–4, 0)
Substitute (–4, 0) into = ? = ?
0 16  ≠
Since (–4, 0) is not a solution of , (–4, 0) is not on the graph.

60. Quadratic Equations and Their Graphs
For any quadratic equation in two variables
all points on its graph are solutions to equation.
all solutions to the equation appear on its graph.
Is the point on the graph of ?
(4, 16)
Substitute = ? = ?
Simplify
16 = 16 
(4, 16) is a solution of ,
(4, 16) is on the graph.

61.  Without graphing, tell whether the point
is on the graph of x2 + y = 2.
(1, 1)
Substitute (1, 1) into
x2 + y = 2.
x2 + y = 2 = ? = ? 
2 = 2
Since (1, 1) is a solution of , (1, 1) is on the graph.
x2 + y = 2

62. 1. In which quadrant is the point with coordinates (7, –2). 2
1. In which quadrant is the point with coordinates (7, –2)? 2. In which quadrant is the point with coordinates (–14, –7) located? 3. Name four ordered pairs. Provide one for each of the quadrants formed by the coordinate plane. Explain. 4. In which quadrant are both coordinates of a point negative? IV
III
III
5. In which quadrant is x-coordinate negative & y-coord. positive? 6. Graph the point C(1, 3). II
7.The scatter plot shows the # of flower orders placed before Valentine’s Day at one shop. Based on this relationship, predict the # of flower orders placed on February 12. -5 5 10
-10
about 45  
8. Evaluate the following if x = and y = -3
2x - 4y – x – y x +-y 8 12 7 62

63. Determine whether each relation is a function. Why?
There are no repeats in the domain. - is a function.
{(3, 0), (4, -6), (-3, 10), (-8, 10)}
no repeats in the domain.
arrows leaving domain don’t repeat.
it is a function. B. X Y 3 4 -3 -8 -6 10
5 in the x pairs up with both 2 and 3 in the y.
one domain pairs with two different ranges.
a repeat in the domain.
A repeat in domain - not a function. C.
Express the relation {(2, 3), (4, 7), (6, 8)} as a table, as a graph, and as a mapping diagram. State whether it is a function.
Express the relation {(1, 3), (2, 4), (1, 5)} as a table, as a graph, and as a mapping diagram. State whether it is a function.

64. Express the relation {(2, 3), (4, 7), (6, 8)} as a table, as a graph, and as a mapping diagram.
x y
Table
Graph x y 2 4 6 3 7 8 2 6 4 3 8 7
it is a function.

65. Express the relation {(1, 3), (2, 4), (1, 5)} as a table, as a graph, and as a mapping diagram.
x y x y 1 3 1 3 2 4 2 4 1 5 5
not a function.

66. Graph the relation on a coordinate plane (-3,3), (1,-2), (1,1), (4,4)
WARM UP
Define:
relation –
domain –
range –
List the domain and range for the relation
relation -> {(1, 2), (3, 5), (8, 4), (6, 2), (3, 1), (7, 4)}
function –
vertical axis –
horizontal axis –
ordered pair -
origin - x y
Complete a table for the given info:
Domain: {-4, -2, 0} Range: {5, 6, 2}
(3,1), (4,-2)
Graph the relation on a coordinate plane
(-3,3), (1,-2), (1,1), (4,4) 66

67. 1-9 A PREVIEW OF GRAPHS AND FUNCTIONS
Graphing Terms:
function - a relationship between input and output
the output depends on the input and there is only one output for every input
vertical axis - the upward axis
represents current car value from the sample graph
horizontal axis - the sideways axis
represents age of car from the sample graph
ordered pair - a pair of numbers used to locate points on a graph
examples from the sample graph are: (2, 8), (1, 11), and (3, 6)
in ordered pairs, the horizontal # always comes first
origin - the point on a graph where all values are zero
Generally, this is the point (0, 0). 67

68. WARM UP x y -4 5 -2 6 2 relation - a set of ordered pairs
domain - the set of first numbers from a set of ordered pairs
range - the set of second numbers from a set of ordered pairs
The following is a relation.
The domain is in blue and the range is in red.
relation -> {(1, 2), (3, 5), (8, 4), (6, 2), (3, 1), (7, 4)}
domain -> {1, 3, 8, 6, 3, 7}
range -> {2, 5, 4, 2, 1, 4}
{1, 3, 6, 7, 8} To “spell” it right,
{1, 2, 4, 5} in order & no repeats.
Complete a table for the given info: x y -4 5 -2 6 2
Domain: {-4, -2, 0} Range: {5, 6, 2} 68

69. Graphing Relations Quadrant II X<0, y>0 Quadrant I
(4,4)
(-3,3)
(1,1)
(3,1)
Origin (0,0)
(1,-2)
(4,-2)
Quadrant IV
X>0, y<0
Quadrant III
X<0, y<0 69

70. Which relation is a function? A. B. C. D.
{(11, –2), (12, –1), (13, 0), (21, 8)}
{(1, –2), (2, –1), (3, 0), (1, 4)}
{(1, –2), (1, –1), (1, 0), (1, 1)}
{(11, –2), (10, –1), (10, 9), (11, 9)}
What is the domain of the function shown in table? A. B. C. D.
{1, 3, 5, 8}
{4}
{7}
{–2, 0, 2}
For f(x) = 3x + 2, find f(x) when x = 7 & x = –4. 70

71. Evaluate the function for the given input values.
For f(x) = 3x + 2, find f(x) when x = 7 & x = –4.
f(x) = 3(x) + 2
f(x) = 3(x) + 2
Substitute
7 for x.
Substitute
–4 for x.
f(7) = 3(7) + 2
f(–4) = 3(–4) + 2 =
= –12 + 2
Simplify.
Simplify.
= 23
= –10
For h(c) = 2c – 1, find h(c) when c = 1 and c = –3.
h(c) = 2c – 1
h(c) = 2c – 1
h(1) = 2(1) – 1
h(–3) = 2(–3) – 1
= 2 – 1
= –6 – 1
= 1
= –7 71

72. Solve each equation or inequality. 3x – 2 = 13 x = 5
(2x + 7)/3 = x + 3 x = -2
-12x > x < 2
7x < 2x – 25 x < -5 72

73. 1. In which quadrant is the point with coordinates (7, –2). 2
1. In which quadrant is the point with coordinates (7, –2)? 2. In which quadrant is the point with coordinates (–14, –7) located? 3. Name four ordered pairs. Provide one for each of the quadrants formed by the coordinate plane. Explain why you selected those you chose. 4. In which quadrant are both coordinates of a point negative? IV
III
III
5. In which quadrant is the x-coordinate negative and the y-coordinate positive? 6. Graph the point C(1, 3). II
7.The scatter plot shows the # of orders placed for flowers before Valentine’s Day at one shop. Based on this relationship, predict the # of flower orders placed on February 12.
about 45 -5 5 10
-10  
2x2 - y = 7
8. Is each equation a linear equation ?
4x = 7 + 2y
yes
x = 12
yes no 73

74. Graphing Linear Equations
Graphing Linear Equations
USING TABLES 74

75. you make it up! TABLE METHOD STEPS:
used to draw the graph of an equation: x y
Create table & find solutions (ordered pairs)
(Substitute value for x. Solve for y.)
3 POINTS FOR LINEAR – 0 , 1 positive #, 1 negative #
5 POINTS FOR NON-LINEAR , 2 positive #s 2 negative #s
you make it up!
2. Plot points on coordinate plane.
Draw a line thru the points, w/ arrows at both ends. 75

76. y = 3(1) – 2 y = 3 – 2 y = 1 y = 3x – 2 x y y = 3(–1) – 2 y = -3 – 2
Graph the linear equation.
y = 3x – 2
1. Draw a table & Choose 3 values for x and solve for ‘y’.
Evaluate for x= ( -1, 0, 1)
y = 3(1) – 2
y = 3 – 2
y = 1 x
y = 3x – 2 y
y = 3(–1) – 2
y = -3 – 2
y = –5 –1 –5 –2  1 1
y = 3(0) – 2
y = 0 – 2
y = –2
2. Plot points.
3. Draw a line through all points to show all ordered pairs that satisfy the equation.
REMEMBER: Draw arrows on both “ends” of the line. 76

77. GRAPHING TIPS
INCHES 10 1 7 4 3 2 6 5 9 8
Use graph paper.
2. Use the grid of the graph.
(Do not re-# or divide grid into smaller segments)
3. Draw small points
4. Use a straight edge 10
DAYS 1 7 4 3 2 6 5 9 8

78. y= -3(1) + 2 y = -3 + 2 y = -1 y = -3x + 2 x y y = -3(–1) + 2
Graph the linear equation.
1. Draw a table & Choose 3 values for x and solve for ‘y’.
Evaluate for x= ( -1, 0, 1)
y= -3(1) + 2
y =
y = -1 x
y = -3x + 2 y
y = -3(–1) + 2
y = 3 + 2
y = 5 –1 5 2  1 -1
y=-3(0) + 2
y = 0 + 2
y=2
2. Plot points.
3. Draw a line through all points to show all ordered pairs that satisfy the equation.
REMEMBER: Draw arrows on both “ends” of the line. 78

79. DON’T HIDE FROM WHAT YOU DONT KNOW HOW TO DO!!!
WHEN DOING HOMEWORK:
Get your notes out. (your personal tutor)
Try each problem using your “personal tutor”.
Look answers up & list left of each problem.
Highlight where you are getting stuck.
Fix your notes in class the next day, by:
comparing them to a friend’s
asking questions during lecture & review
take better notes during lecture

80. TRY THESE x y Graph the following using table method: y = | x | + 1
Create table & find solutions (ordered pairs)
(Substitute value for x. Solve for y.)
3 POINTS FOR LINEAR
0 , 1 positive #, 1 negative #
5 POINTS FOR NON-LINEAR
0 , 2 positive #s 2 negative #s
y = -x2 - 5
y =–2x + 3
2. Plot points on coordinate plane.
Draw a line thru the points, w/ arrows at both ends.

81. y = | x | + 1. x y –2 3 –1 2 1 y= | 1 | + 1 y = 1 + 1 y = 2
Graph the linear equation.
1. Draw a table & Choose 3 values for x and solve for ‘y’.
Opens up
Positive x y –2 3 –1 2 1
y= | 1 | + 1
y = 1 + 1
y = 2 x
y = |x | + 1 y
y = |–1 | + 1
y = 1 + 1
y = 2 –1 2 y 1 1 2 4
y= | 0 | + 1
y = 0 + 1
y=1 2 x
2. Plot points. –2 2
3. Draw a line through all points to show all ordered pairs that satisfy the equation.
REMEMBER: Draw arrows on both “ends” of the line.
USE AT LEAST 5 POINTS FOR NON-LINEAR EQUATIONS- pick at least 2 negative #s 81

82. The highest or lowest point on the parabola is called the ver tex.
SYMMETRY
The highest or lowest point on the
parabola is called the ver tex.
axis of symmetry for the parabola is the vertical line through the vertex. 82

83. y = -x2 - 5 x y –3 –14 –2 –9 –1 –6 –5 1 2 3 y = -x2- 5 y= -(0 )2 - 5
Graph the linear equation. x y –3
–14 –2 –9 –1 –6 –5 1 2 3
1. Draw a table & Choose 3 values for x and solve for ‘y’.
Opens down
Negative
y = -x2- 5
y= -(0 )2 - 5
y =
y= -5
y= -( 1 )2 - 5
y = -1 – 5
y = -6
y = -(–1 )2 - 5
y =
y = -6 x y –1 -6 -5 1 -6
2. Plot points.
3. Draw through all points to show all ordered pairs that satisfy the equation.
REMEMBER: Draw arrows on both “ends”.
USE AT LEAST 5 POINTS FOR NON-LINEAR EQUATIONS- pick at least 2 negative #s 83

84. x y –2 7 –1 5 3 1 2 y= -2(1) + 3 y = -2 + 3 y = 1 y = -2x + 3 x y
Graph the linear equation.
y =–2x + 3
1. Draw a table & Choose 3 values for x and solve for ‘y’.
y= -2(1) + 3
y =
y = 1 x y –2 7 –1 5 3 1 2 x
y = -2x + 3 y
y = -2(–1) + 3
y = 2 + 3
y = 5 –1 5 y 3 8 1 1
y=-2(0) + 3
y = 0 + 3
y=3 4 x 4 4 8
2. Plot points. –4
3. Draw a line through all points to show all ordered pairs that satisfy the equation.
REMEMBER: Draw arrows on both “ends” of the line. 84

85. Relations and Functionsx y -4 5 -2 6 2 1 7 3 -5
Example:
The vertical line test can be used to determine if the relation is a function. Use a straight edge moved slowly across the graph to see if two point touch the edge at the same time. This is a true test for all functions.
A Function is a relation in which each element of the domain is paired with exactly one element in the range. 85

86. H G VERTICAL LINES Graph of x = # is a vertical line through (# , 0).
Vertical line x=-2 crosses
the x axis at (-2,0)
(4,6) H
x=4 G
(4,0)
Vertical Lines domain repeats. They are never functions.
What is the equation for line HG ? 86

87. Vertical Line Test Determine whether x - 4y = 12 is a function.
METHOD 1
Graph it.
Is it a linear equation?
Yes.
Is the linear equation a vertical line?
No.
Remember:
All non-vertical linear equations are functions.
This is a function.
Try to draw a vertical line through any two points.
Cannot draw a vertical line through any two points.
It passes the test. This relation is a function. 87

88. Complete the tables Warm Up y= 2x - 1 -x – 1 = y 3x – 2y = 6 x y 2 -1-1 3 -3 -1 -1 -3 -3
-4.5 88

89. Graph: y = x + 1
Table of values 89

90. Practice for Examples 3 and 45.
Gr zph the equation 1. 2x = – y +
ANSWER
Explain why no other combo
of shawls and scarves are
possible in Example 4. 6.
ANSWER
The numbers of shawls and
scarves must be whole numbers.

91. 5-4: Graphing Linear Equations
EXAMPLE 3: Graph the equation 2x + 5y = 10.
solve equation for y.
2x + 5y = 10
-2x x
5y = x
5 5
Plot the points and draw a line
2/5 x,
choosing values of x divisible by five will make the problem easier.
y-values will be whole numbers make a table.

92. Two-variable equation solution is a set of ordered pairs.
Sometimes, need to solve for y first, then set up a table of values.
Solve 8x + 4y = 24 if the domain is {-2, 0, 5, 8}.
Two-variable equation solution is a set of ordered pairs.
You are given the domain (x-value.) in each ordered pair, Need the second number, the range (y-value.) in each ordered pair,
To find y, solve the equation for y. x -2 5 8
6 - 2x
6 - 2(-2)
6 - 2(0)
6 - 2(5)
6 - 2(8) y 10 6 -4
-10
(x, y)
(-2, 10)
(0, 6)
(5, -4)
(8, -10)
Now make a table.
8x + 4y = 24
-8x x
4y = x
4 4
y = 6 - 2x
This is your solution.
{(-2, 10), (0, 6), (5, -4), (8, -10)}

93. Finding Solutions of an Equation
List four solutions of the equation 5x + y =
12.
Solve for y and then find y-values for several x-values.
1 Solve the equation for y. 12 y 5x = + 12 y 5x = –
2 Find y-values for at least x-values. 2 1 –1 x 12 y 5x = – ( )
x, y
2, 2
1, 7
0, 12
–1, 17 5 17 7
Four solutions are ( ).
2, 2 )
1, 7 ,
0, 12 ,
–1, 17 ,
and

94. SOME FACTS ABOUT FUNCTIONS TO THINK ABOUT
1. Graph y = -5x + 3
y + x = 7
2. Define function. Identify if both graphs are functions
3. What are the synonyms for domain and range?
4. Define :
Solutions set for a linear equation
Linear equation
5. Determine whether (0, 3) (-3, 6) (4, 2) & (3, 0) are solutions to x + y = 3
6. The table shows the # of cookies in a jar from the time since they were
baked. Graph a scatter plot
using the given data.
SOME FACTS ABOUT FUNCTIONS TO THINK ABOUT
☻Functions are nearly the same concept as graphing. ☻Terminology is different & and few additional rules. ☻Processes are the same. ☻Future math classes, & standardized tests will refer to functions. This is important.

95. y= -5(1) + 3 y = -5 + 3 y = -2 y = -5x + 3 x y y = -5(–1) + 3
Graph y = -5x + 3. Use a table.
1. Draw a table & Choose 3 values for x and solve for ‘y’.
y= -5(1) + 3
y =
y = -2 x
y = -5x + 3 y
y = -5(–1) + 3
y = 5 + 3
y = 8 –1 8  3 1 -2
y=-5(0) + 3
y = 0 + 3
y=3
2. Plot points.
3. Draw a line through all points to show all ordered pairs that satisfy the equation.
REMEMBER: Draw arrows on both “ends” of the line.

96. y +(1) = 7 -1 -1 y = 6 y + x = 7 x y y +(–1) = 7 +1 +1 –1 8 y = 8 7 1
Graph y + x = 7. Use a table.
1. Draw a table & Choose 3 values for x and solve for ‘y’.
y +(1) = 7
y = 6 x
y + x = 7 y
y +(–1) = 7
y = 8 –1 8  7 1 6
y +(0) = 7
y = 7
2. Plot points.
3. Draw a line through all points to show all ordered pairs that satisfy the equation.
REMEMBER: Draw arrows on both “ends” of the line.

97. linear. not SOLUTION SET: (the list of solutions)
For Linear Equations, ORDERED PAIRS are solutions
GRAPH of a Linear Equation is the drawing that represents the solution set.
Linear Equation(equation whose graph forms a line.
linear.
not
SUBSTITUTION & SOLVE (determine if equation is true)
x + y = 3 substitute (0, 3)
x + y = 3 substitute (-3, 6)
x + y = 3 substitute (4, 2)
x + y = 3 substitute (3, 0)
*If they are true, then the coordinates are an ordered pair
of the linear equation.
TRUE
TRUE
FALSE
TRUE 97

98. Graphing a Scatter Plot from Given Data
The table shows the number of cookies in a jar from the time since they were baked. Graph a scatter plot using the given data.
Use the table to make ordered pairs for the scatter plot.
The x-value represents the time since the
cookies were baked and the y-value
represents the number of cookies left in
the jar.
Plot the ordered pairs.

99. A. B. C. D. Which graph matches the values from the table?
Graph the equations.
y = |x| + 2
y = 4x 1 y x = 2 –
y = 2x + 1

100. 1 + 2 = 3 0 + 2 = 2 –1 1 |x| + 2 = y x (x, y) (–1, 3) (0, 2) (1, 3)
Graph the equation.
y = |x| + 2
Opens up
Positive
Step 1 Choose several values of x and generate ordered pairs. 
1 + 2 = 3
0 + 2 = 2 –1 1
|x| + 2 = y x
(x, y)
(–1, 3)
(0, 2)
(1, 3) 
Step 2 Plot enough points to see a pattern.
100

101. x y = 4x (x, y) 1 y = 4(1) = 4 (1, 4) y = 4(0) = 0 (0, 0) –1
Graph y = 4x. Use slope intercept method
Step 1 Choose three values of x and generate ordered pairs  x
y = 4x
(x, y) 1
y = 4(1) = 4
(1, 4)
y = 4(0) = 0
(0, 0) –1
y = 4(–1) = –4
(–1, –4)
Step 2 Plot the points and connect them with a straight line.
No vertical line will intersect this graph more than once. So y = 4x describes a function.

102. x y = 2x + 1 (x, y) y = 2(1) + 1 = 3 (1, 3) 1 y = 2(0) + 1 = 1 (0, 1)
Graphing Linear Equations
Graph y = 2x + 1
Step 1 Choose three values of x and generate ordered pairs.  x
y = 2x + 1
(x, y)
y = 2(1) + 1 = 3
(1, 3) 1
y = 2(0) + 1 = 1
(0, 1) –1
y = 2(–1) + 1 = –1
(–1, –1)
Step 2 Plot the points and connect them with a straight line.
No vertical line will intersect this graph more than once. So y = 2x + 1 describes a function.

103. x – 2 2 4 y – 1 1 Graph the equation x – 2y = 2.
STEP 1 Solve the equation for y, then make a table by
choosing a few values for x and finding the values of y. 2 2y x = – 2 2y x = – + 1 y x = 2 – x
– 2 2 4 y
– 1 1
STEP 2 Plot the points. Draw a line through them.
103

104. WARM UP
Create a table & graph f(x) = 6 - 2x if the domain is {-2, 0, 5, 8}.
Graph the equation y = x. Evaluate for x= {-10,-5,0,5,10}
Graph y= 3x -4. Use a table.
Determine whether each relation is a function. Why?
A. {(2, 3), (3, 0), (5, 2), (-1, -2), (4, 1)} B. X Y C. 3 4 -3 -8 -6 10

105.  Given the domain (x-value.) in each ordered pair,
Create a table and graph f(x) = 6 - 2x if the domain is {-2, 0, 5, 8}.
Given the domain (x-value.) in each ordered pair,
2. To find y, solve the equation for y.
3. Make a table. x -2 5 8
6 - 2x
6 - 2(-2)
6 - 2(0)
6 - 2(5)
6 - 2(8) y 10 6 -4
-10
(x, y)
(-2, 10)
(0, 6)
(5, -4)
(8, -10)
4. Plot points.
5. Draw a line 

106. Graphing Linear Equations
Graph the equation y = x. Evaluate for x= {-10,-5,0,5,10}
Plot the points and draw a line

107. Graphing Linear Equations
Use a table and graph the equation y = 3x - 4.
1. Substitute 3 values in the equation
2 Solve for y.
5. Draw a line through all points.
Arrows are on both ends!
3. Write the ordered pairs.
4. Plot points on a graph. x -1 1 2 3
3x - 4
3(-1) - 4
3(0) - 4
3(1) - 4
3(2) - 4
3(3) - 4 y -7 -4 -1 2 5
(x, y)
(-1,-7)
(0, -4)
(1, -1)
(2, 2)
(3, 5)
Is (6,0) a solution to this equation?)
107

108. Determine whether each relation is a function. Why?
There are no repeats in the domain. - is a function.
{(3, 0), (4, -6), (-3, 10), (-8, 10)}
no repeats in the domain.
arrows leaving domain don’t repeat.
it is a function. B. X Y 3 4 -3 -8 -6 10
5 in the x pairs up with both 2 and 3 in the y.
one domain pairs with two different ranges.
a repeat in the domain.
A repeat in domain - not a function. C.

109. SOME FACTS ABOUT FUNCTIONS TO THINK ABOUT
1. Graph f(x) = -5x + 3
y + x = 7
2. Define function. Identify if both graphs are functions
3. What are the synonyms for domain and range?
4. Define :
Solutions set for a linear equation
Linear equation
5. Determine whether (0, 3) (-3, 6) (4, 2) & (3, 0) are solutions to x + y = 3
6. The table shows the # of cookies in a jar from the time since they were
baked. Graph a scatter plot
using the given data.
SOME FACTS ABOUT FUNCTIONS TO THINK ABOUT
☻Functions are nearly the same concept as graphing. ☻Terminology is different & and few additional rules. ☻Processes are the same. ☻Future math classes, & standardized tests will refer to functions. This is important.

110. y= -5(1) + 3 y = -5 + 3 y = -2 F(x) = -5x + 3 x y y = -5(–1) + 3
Graph f(x) = -5x + 3. Use a table.
1. Draw a table & Choose 3 values for x and solve for ‘y’.
y= -5(1) + 3
y =
y = -2 x
F(x) = -5x + 3 y
y = -5(–1) + 3
y = 5 + 3
y = 8 –1 8  3 1 -2
y=-5(0) + 3
y = 0 + 3
y=3
2. Plot points.
3. Draw a line through all points to show all ordered pairs that satisfy the equation.
REMEMBER: Draw arrows on both “ends” of the line.

111. y +(1) = 7 -1 -1 y = 6 y + x = 7 x y y +(–1) = 7 +1 +1 –1 8 y = 8 7 1
Graph y + x = 7. Use a table.
1. Draw a table & Choose 3 values for x and solve for ‘y’.
y +(1) = 7
y = 6 x
y + x = 7 y
y +(–1) = 7
y = 8 –1 8  7 1 6
y +(0) = 7
y = 7
2. Plot points.
3. Draw a line through all points to show all ordered pairs that satisfy the equation.
REMEMBER: Draw arrows on both “ends” of the line.

112. linear. not SOLUTION SET: (the list of solutions)
For Linear Equations, ORDERED PAIRS are solutions
GRAPH of a Linear Equation is the drawing that represents the solution set.
Linear Equation(equation whose graph forms a line.
linear.
not
SUBSTITUTION & SOLVE (determine if equation is true)
x + y = 3 substitute (0, 3)
x + y = 3 substitute (-3, 6)
x + y = 3 substitute (4, 2)
x + y = 3 substitute (3, 0)
*If they are true, then the coordinates are an ordered pair
of the linear equation.
TRUE
TRUE
FALSE
TRUE
112

113. Graphing a Scatter Plot from Given Data
The table shows the number of cookies in a jar from the time since they were baked. Graph a scatter plot using the given data.
Use the table to make ordered pairs for the scatter plot.
The x-value represents the time since the
cookies were baked and the y-value
represents the number of cookies left in
the jar.
Plot the ordered pairs.

114. Vertical Line
Test

115. Example: Vertical Line Test
Vertical Line Test: Use vertical line test to determine which of the relations are functions. y
x = | y – 2| y
x = 2y – 1 x x
passes is a function.
does not pass not a function.
y 2 = x x y
y = x 2 x y
x 2 + y 2 = 1 x y
2 points of intersection
1 point of intersection
2 points of intersection
Example: Vertical Line Test
Vertical Line Test
115

116. Vertical Line Test Determine whether x - 4y = 12 is a function.
METHOD 1
Graph it.
Is it a linear equation?
Yes.
Is the linear equation a vertical line?
No.
Remember:
All non-vertical linear equations are functions.
This is a function.
Try to draw a vertical line through any two points.
Cannot draw a vertical line through any two points.
It passes the test. This relation is a function.
116

117. Graph y = x + 6. If the domain is {-4, -3, -1, 2, 4}.
This is
a function
1. Make table with given x-values.
2.Determine the y-values.
3.Write solutions as ordered pairs.
Then plot the points.
The solution set is:
{(-4, 2), (-3, 3), (-1, 5), (2, 8), (4, 10)}.
WHERE your warm up equations all functions?
117

118. 1. Choose several values of x & generate ordered pairs.
Graph each equation Then tell whether the equation represents a function.
y = |x – 1|
1. Choose several values of x & generate ordered pairs.
2. Plot enough points.
(2, 1)
(–1, 2)
(0, 1)
(1, 0)
(x, y)
y = |x – 1| x
2 = |–1 – 1|
1 = |0 – 1|
0 = |1 – 1|
1 = |2 – 1| –1 1 2 
3. Points form a V-shaped graph. Draw two rays
4. Use the vertical line test on the graph.
No vertical line will intersect the graph more than once. The equation y = |x – 1| represents a function.
118

119. Step 1 Choose several values of x and generate ordered pairs.
Graph the equation. Then tell whether the equation represents a function.
y = 3x – 2
Step 1 Choose several values of x and generate ordered pairs.
Step 2 Plot enough points to see a pattern.  
3(–1) – 2 = –5 –1 1
3x – 2 = y x
(x, y)
(–1, –5)
(0, –2)
(1, 1)
3(0) – 2 = –2
3(1) – 2 = 1
Step 3 Draw a line through all the points. Remember arrowheads on both “ends” of the line.
Step 4 Use the vertical line test on the graph.
No vertical line will intersect the graph more than once. The equation y = 3x – 2 represents a function.
119

120. TRY THESE
Graph each linear equation. Then tell whether it represents a function.
1. 2y + x = 6
2. 3y = 12
Yes, it is a function.
Yes, it is a function.
Graph each equation. Then tell whether the equation represents a function.
f(x) = 3x – 2
–3x + 2 = y
f(x) = |x| + 2

121. Step 2 Plot enough points to see a pattern.
Graph each equation. Then tell whether the equation represents a function.
f(x) = 3x – 2
Step 2 Plot enough points to see a pattern.
Step 1 Choose several values of x and generate ordered pairs.
3(–1) – 2 = –5 –1 1
3x – 2 = y x
(x, y)
(–1, –5)
(0, –2)
(1, 1)
3(0) – 2 = –2
3(1) – 2 = 1 
No vertical line will intersect the graph more than once. The equation y = 3x – 2 represents a function.
121

122. Step 1 Choose several values of x and generate ordered pairs.
Graph each equation. Then tell whether the equation represents a function.
–3x + 2 = y
Step 1 Choose several values of x and generate ordered pairs.
Step 2 Plot enough points to see a pattern.
–3(–1) + 2 = 5
–3(0) + 2 = 2
–3(1) + 2 =–1 –1 1
–3x + 2 = y x
(x, y)
(–1, 5)
(0, 2)
(1, –1) 
Step 3 The points appear to form a line. Draw a line through all the points to show all the ordered pairs that satisfy the function. Draw arrowheads on both “ends” of the line.
Step 4 Use the vertical line test on the graph.
No vertical line will intersect the graph more than once. The equation –3x + 2 = y represents a function.
122

123. 1. Choose several values of x & generate ordered pairs. f(x) = |x| + 2
Graph each equation. Then tell whether the equation represents a function.
1. Choose several values of x & generate ordered pairs.
f(x) = |x| + 2
1 + 2 = 3
0 + 2 = 2 –1 1
|x| + 2 = y x
(x, y)
(–1, 3)
(0, 2)
(1, 3)
2. Plot 3 points min.
3. Points appear to form a V-shaped graph. Draw two rays from (0, 2) to show all the ordered pairs that satisfy the function. Draw arrowheads on the end of each ray.  
4. Use vertical line test on the graph.
No vertical line will intersect the graph more than once. a function.
When choosing values of x, be sure to choose both positive and negative values.
Helpful Hint
123

124. Evaluate the functions:
Use functional notation with non-linear functions also.
The process is the same. Need to remember how to work with exponents or variables in denominators.
Evaluate the functions:
f(x) = 2x + 3
f(2) =
f(0) =
f(-3) =
f(x) = 3 – x
f(2) =
f(-1) =
f(-3) =
f(x)= - x2
f(3) =
f(-2) =
124

125. Evaluate the function for the given input values.
For y = 4x2, find y when x = 7 & x = –4.
y = 3(x)2
y = 3(x)2
Substitute
7 for x.
Substitute
–4 for x.
y = 3(7)2
y = 3(–4)2
y = 3(49)
Simplify.
y = 3(16)
Simplify. 7
147 -4 48
( , )
( , )
y= 147
y = 48
Find the output for each input: 1,-3
y = 3x+ 2
y = 3x + 2
y = 3(1) + 2
y = 3(–3) + 2
y= 3 +2
y= –9 + 2 1 5 -3 -7
y= 5
( , )
y= –7
( , )
125

126. 1-9 A PREVIEW OF GRAPHS AND FUNCTIONS
Here is a labeled diagram of a graph.
Function - the equation being graphed by the purple line
Ordered Pairs
(4, 8)
(1, 6)
Vertical Axis
(5, 3)
(3, 2)
(2, 1)
Origin
Horizontal Axis
126

127. f(x) = |x+1| + 2 You pick any 5 values. Pick some negative #s
Create a table and then graph each of the following and state whether it is a function.
f (x) = x2 – 2x – for f(2), f(-2), f(0), f(4) and f(4)
y=5x You pick any 3 values
y =|x| You pick any 5 values. Pick some negative #s
f(x) = |x+1| You pick any 5 values. Pick some negative #s

128. Function
Notation
with exponents
& other variables
128

129. Let f (x) = x2 – 3x – 1. Find f (–2).
TRY THESE
Let f (x) = x2 – 3x – 1. Find f (–2).
Let f (x) = 4x – x2. Find f (x + 2).
For , find h(r) when r = 600 & r = –12.
For g(t) = 1.5t – 5, find g(t) when t = 6 & t = –2.
For g(t) = find g(t) when t = 20 & when t = –12.
For f(x) = 6x – 1, find f(x) when x = 3.5 & when x = –5.
If h(z) = z2 - 4z + 9, find each value. h(-3) h(5c) 5[h(c)]

130. Let f (x) = x2 – 3x – 1. Find f (–2).
Functions can be named with any letter; f, g, and h are the most common. You read f(6) as “f of 6,” and g(2) as “g of 2.”
Reading Math
To evaluate a function f (x) at x = a, substitute the specified value a for x into the given function.
Let f (x) = x2 – 3x – 1. Find f (–2).
f (x) = x2 – 3x – 1
f (–2) = (–2)2 – 3(–2) – Substitute –2 for x.
f (–2) = – Simplify.
f (–2) = The value of f at –2 is 9.
Evaluating Functions
130

131. Example: Evaluating Functions
Example: Let f (x) = 4x – x2. Find f (x + 2).
f (x) = 4x – x2
f (x + 2) = 4(x + 2) – (x + 2) Substitute x + 2 for x.
f (x + 2) = 4x + 8 – (x2 + 4x + 4) Expand (x + 2)2.
f (x + 2) = 4x + 8 – x2 – 4x – Distribute –1.
f (x + 2) = 4 – x The value of f at x is 4 – x2.
Example: Evaluating Functions
131

132. variable used will not always be x and y.
Equations as Relations
variable used will not always be x and y.
Sometimes it is obvious which variable is the domain and which is the range.
Other times, how the equations are written will determine which variable stands for the domain and which variable stands for the range.
On any test I give you, x will be the domain and y will be the range.
Some problems in the book expect you to assume the domain is the letter closest to “A” in the alphabet. The letter furthest from “A” in the alphabet is the range.
Unfortunately, this little rule does not always work.
Functions can be named with any letter; f, g, and h are the most common. You read f(6) as “f of 6,” and g(2) as “g of 2.”
Reading Math
132

133. Evaluate the function for the given input values.
For , find h(r) when r = 600 & r = –12.
= 202
= –2
For g(t) = 1.5t – 5, find g(t) when t = 6 & t = –2.
g(t) = 1.5t – 5
g(t) = 1.5t – 5
g(6) = 1.5(6) – 5
g(–2) = 1.5(–2) – 5
= 9 – 5
= –3 – 5
= 4
= –8
133

134. Lesson Quiz: Part III
Evaluate each function for the given input values.
4. For g(t) = find g(t) when t = 20 and when t = –12.
g(20) = 2
g(–12) = –6
5. For f(x) = 6x – 1, find f(x) when x = 3.5 and when x = –5.
f(3.5) = 20
f(–5) = –31

135. If h(z) = z2 - 4z + 9, find each value. A. h(-3)
B. h(5c)
C. 5[h(c)]
h(-3) = (-3)2 - 4(-3) + 9 =
= 30
h(5c) = (5c)2 - 4(5c) + 9
= 25c2 - 20c + 9
No like terms to combine, so that is the answer.
First find h(c), then multiply by the whole thing by 5.
h(c) = (c)2 - 4(c) + 9
= c2 - 4c + 9
No like terms to combine, so multiply by 5.
5[h(c)] = 5c2 - 20c + 45
135

136. plug in 4 where there is an x.
Function Notation
f(x) is another way to write “y” in functional notation
y is the dependent variable …it depends on x…
SO y is a function of x…
so, y = f(x).
equation:
y = 3x + 7
functional notation:
f(x) = 3x + 7
Find the value of a function for a given domain, (same as finding out the y-value for a given x-value using the table method.)
EX : f(x) = 3x means: find “f of 4” or f(4)
plug in 4 where there is an x.
f(4) = 3(4) + 7
f(4) =
f(4) = 19
136

137. Graph
137

138. Let’s Graph Example 4
138

139. Let’s Graph Example 5
139

140. Let’s Graph Example 8
140

141. Let’s Graph Example 1

142. Let’s Graph Example 2

143. Let’s Graph Example 3
Timer

144. 4-1 Graphing Relationships
Lesson Quiz: Part II
2. A pet store is selling puppies for $50 each. It has 8 puppies to sell. Sketch a graph for this situation.

145. 4-3 Writing and Graphing Functions
Lesson Quiz: Part I
1. Graph y = |x + 3|.

146. 4-3 Writing and Graphing Functions
Lesson Quiz: Part III
Evaluate each function for the given input values.
4. For g(t) = find g(t) when t = 20 and when t = –12.
g(20) = 2
g(–12) = –6
5. For f(x) = 6x – 1, find f(x) when x = 3.5 and when x = –5.
f(3.5) = 20
f(–5) = –31

147. 4-3 Writing and Graphing Functions
Lesson Quiz: Part Il
Identify the independent and dependent variables. Write a rule in function notation for each situation.
2. A buffet charges $8.95 per person.
independent: number of people
dependent: cost
f(p) = 8.95p
3. A moving company charges $130 for weekly truck rental plus $1.50 per mile.
independent: miles
dependent: cost
f(m) = m

148. MB Solve the equation. 5. 2x – 17 = 13 ANSWER 15 6. 4 = 15x – 11 1
6. 4 =
15x – 11 1
ANSWER x – + 30 =
ANSWER 5
8. 7x – 3x = 16
ANSWER 4 –
ANSWER – 2 1 = x + 5x –
10. 7x + 17 – 25 = –
ANSWER 6

149. Without graphing, tell whether each point is on the graph of 6x – 2y = 8.
3. (1, 1) no
4. (3, 5)
yes
5. The cost of a can of iced-tea mix at Save More Grocery is $4.75. The function f(x) = 4.75x gives the cost of x cans of iced-tea mix. Graph this function and give its domain and range.
D: {0, 1, 2, 3, …}
R: {0, 4.75, 9.50, 14.25, …}
149

151. Create a table of values for each equation, and graph it on grid paper:1. 2. 3. 4.
151

152. Warm-Up Exercises
Evaluate the expression. 1. 2. 23 – 3 5 3. 2 ( ) 4 2
ANSWER 23 –
Tell whether the number is a solution of the equation. 1. 2. 2 4 3x = – 2; 14 6 4a 5;
ANSWER
yes no

153. WARM UP
Write four solutions for each equation. Write the solutions as ordered pairs.
y = 2x – 4
y = x/2 – 3
x + y = 12
Determine whether each relation is linear.
x – y = 32
25 – y2 = x2
153