1. GRAPH OF
FUNCTIONS

2. • A relation can be described as a graph
a. A = {(-5, -5), (-3, -3), (-1, -1), (1, 1),(3, 3), (5, 5)}
Since the domain is limited to
the set D = {-5, -3, -1, 1, 3, 5} ,
points should not be connected

3. An Equation can also be described as a graph
b. y = 2x + 1
x
y
-2
-1
0
1
2

4. An Equation can also be described as a graph
b. y = 2x + 1
x
y
-2
-3
-1
0
1
2

5. An Equation can also be described as a graph
b. y = 2x + 1
x
y
-2
-3
-1
-1
0
1
2

6. An Equation can also be described as a graph
b. y = 2x + 1
x
y
-2
-3
-1
-1
0
1
1
2

7. An Equation can also be described as a graph
b. y = 2x + 1
x
y
-2
-3
-1
-1
0
1
1
3
2

8. An Equation can also be described as a graph
b. y = 2x + 1
x
y
-2
-3
-1
-1
0
1
1
3
2
5

9. An Equation can also be described as a graph
b. y = 2x + 1
x
y
-2
-3
-1
-1
0
1
1
3
No domain is specified
when a function is
defined
2
5

10. c. y2 = x
x
y
0

11. c. y2 = x
x
y
0
1

12. c. y2 = x
x
y
0
1
4

13. c. y2 = x
x
y
0
1
4
9

14. c. y2 = x
x
y
0
0
1
4
9
16

15. c. y2 = x
x
y
0
0
1
+1
4
9

16. c. y2 = x
x
y
0
0
1 4
+1 +2
9

17. c. y2 = x
x
y
0
0
1 4 9
+1 +2 +3

18. c. y2 = x
x
y
0
0
1 4 9
+1 +2 +3
The domain of this kind
of relation is { x x > 0 }

19. The Vertical Line Test
• A graph of a relation is a function if any vertical line
drawn passing through the graph intersects it at
exactly one point.
 Determine which of the following graphs of relation represents a
function.

20. • Constant Functions

A constant function C consists of a single real number k in its range for
all real numbers x in its domain.

21. IDENTITY
FUNCTION
I(x) = x
If the domain is specified to be the set of
all real numbers, the range of the identity
function is also the set of all real numbers

22. • Some points on the graph of an I(x) = x are
(-2, -2), (-1, -1), (0, 0),(1, 1), (2, 2)

23. POLYNOMIAL
FUNCTIONS
A constant function is a polynomial function
of the degree 0. If a polynomial function is of
the first degree, then it is called a linear
function

24. Polynomial Functions:
Linear Function
f (x) = mx + b
Note: f(x) = y

25. • Draw the graph of a linear function
f(x) = -2x + 5
x
f(x
)
-1
0
1
2
3

26. • Draw the graph of a linear function
f(x) = -2x + 5
x
-1
f(x
)
7
0
1
2
3

27. • Draw the graph of a linear function
f(x) = -2x + 5
x
-1
0
f(x
)
7
5
1
2
3

28. • Draw the graph of a linear function
f(x) = -2x + 5
x
-1
0
1
f(x
)
7
5
3
2
3

29. • Draw the graph of a linear function
f(x) = -2x + 5
x
-1
0
1
2
f(x
)
7
5
3
1
3

30. • Draw the graph of a linear function
f(x) = -2x + 5
x
-1
0
1
2
3
f(x
)
7
5
3
1
-1
The domain is x x is
a real number and it
follows that the range
is y y is a real number

31. If a polynomial function is of the second
degree, then it is called a quadratic
function

32. Polynomial Functions:
Quadratic Function
f (x) = ax2 + bx + c

33. • Draw the graph of the quadratic equation
g(x) = x2
x
g(x
)
-2
-1
0
1
2

34. • Draw the graph of the quadratic equation
g(x) = x2
x
-2
g(x
)
4
-1
0
1
2

35. • Draw the graph of the quadratic equation
g(x) = x2
x
-2
-1
g(x
)
4
1
0
1
2

36. • Draw the graph of the quadratic equation
g(x) = x2
x
-2
-1
0
g(x
)
4
1
0
1
2

37. • Draw the graph of the quadratic equation
g(x) = x2
x
-2
-1
0
1
g(x
)
4
1
0
1
2

38. • Draw the graph of the quadratic equation
g(x) = x2
x
-2
-1
0
1
2
g(x
)
4
1
0
1
4
A quadratic function
is a parabola.
The range for both
function is {y ǀ y > 0
}

39. Absolute Value Functions:
f (x) = x

40. The domain of an absolute
value function is the set of real
numbers and the range is {f(x)
f(x) > 0 }

41. Example: In one Cartesian
plane, draw the graph and
determine the domain and
range of each function.
y= x
a. y = x + 2
Simply shift to the
left
b. y = x - 2
Simply shift to the
right
y= x+2
The domain for both
function is the set of all
real numbers
y= x-2
The range for both
function is {y ǀ y > 0
}

42. Example: In one Cartesian
plane, draw the graph and
determine the domain and
range of each function.
y= x+2
y= x
a. y = x + 2
b. y = x - 2
The absolute sign does
not affect the constant.
y= x -2
The domain for both
function is the set of all
real numbers
The range for both
function is {y ǀ y > -2
}

43. • Draw the graph of each function. Determine its
domain and range.
1. y = 7
6. y = 3x – 1
2. y = -5
7. y = 2x2
3. y = 2x + 3
8.
4. y = ǀ x + 3 ǀ
9. y = ǀ x – 3 ǀ
5. y = ǀxǀ - 4
y = x2 + 2
10. y = ǀxǀ + 4