2.1 – Quadratic Functions

In this section, you will learn to

analyze graphs of quadratic functions

write quadratic functions in standard form and sketch its graphs

solve real-life problems

Definition of a Polynomial Function:

1 2 21 2 2 1 0....

is a polynomial function of with degree .

n n nn n nf x a x a x a x a x a x a

x n

Definition of a Quadratic Function:

a) Axis of symmetry: the line where the parabola is symmetric

b) Vertex: The point where the axis of symmetry intersects the parabola

2 , 0f x ax bx c a

2x

2, 2

1 2 3 4 5 6-1-2

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

Definition of a Quadratic Function:

c) Upward or Downward: If the leading coefficient is positive (a>0) , the parabola opens upward.

2 , 0f x ax bx c a

1 2 3 4-1-2-3-4

1

2

3

4

-1

-2

-3

-4

x

y

22 1

2 0

f x x

Definition of a Quadratic Function:

c) Upward or Downward: If the leading coefficient is negative (a<0) , the parabola opens downward.

2 , 0f x ax bx c a

22 1

2 0

f x x

1 2 3 4-1-2-3-4

1

2

3

4

-1

-2

-3

-4

x

y

Definition of a Quadratic Function:

d) Minimum or Maximum: If the parabola opens upward, the vertex has a minimum value. If the parabola opens downward, the vertex has a maximum value.

2 , 0f x ax bx c a

1 2 3 4-1-2-3-4

1

2

3

4

-1

-2

x

y

1 2 3 4-1-2-3-4

1

2

-1

-2

-3

-4

x

y

Minimum

Maximum

Standard Form of a Quadratic Function:

a) Vertex:b) Axis of Symmetry: c) Vertex:

Therefore,

* To write an equation in standard form, you need to complete the

square.

2, 0f x a x h k a

2 2

b bh k f f h

a a

, ,2 2

b bh k f

a a

,h k

x h

Identify the vertex and axis of symmetry for

There are two methods to identify the vertex and the axis of symmetry.

Method 1:

24 2 1f x x x

2 1

2 2 4 4

bh

a

21 1 1 3

4 2 14 4 4 4

k f

1Axis of Symmetry :

4x

Method 2: Complete the Square

24 2 1f x x x

21 4 2f x x x

2 11 4

2f x x x

2 1

1 42

1

4 16

1f x x x

2

3 14

4 4f x x

1 3 1, , Axis of Symmetry:

4 4 4h k x

2

1 34

4 4f x x

Complete the Square:

212 1

2f x x x

211 2

2f x x x

211 4

2f x x x

211 4

22 4f x x x

213 2

2f x x 21

2 32

f x x

, 2, 3 Axis of Symmetry: 2h k x

Method 2: Complete the Square

22 3 1f x x x

21 2 3f x x x

2 31 2

2f x x x

2 31 2

2

9

8

9

16f x x x

2

1 32

8 4f x x

3 1 3, , Axis of Symmetry:

4 8 4h k x

2

3 12

4 8f x x

Identify the vertex and zeros to graph:

Vertex:

Zeros:

22 8 3f x x x

8

22 2

h

28 8 4 2 3

2 2x

22 2 2 8 2 3 5k f

, 2,5h k

8 2 10 4 10

4 2

Identify the vertex and zeros to graph:

Vertex: Zeros:

22 8 3f x x x

2,54 10

0.42, 3.582

1 2 3 4 5-1

1

2

3

4

5

6

-1

-2

-3

-4

-5

x

y

Find the quadratic equation in standard form:

Find the standard form of the equation of

the parabola whose vertex is   2, 3   and

passes through the point  3,5 .

Find the quadratic equation in standard form:

Vertex:Point:

, 2, 3h k , 3,5x y

2f x a x h k

25 3 2 3a

5 25 3a 8 25a

28 82 3

25 25a f x x

Real-Life Example:

A baseball is hit at a point 5 feet above the ground

at a velocity of 100 feet per second and at an angle

of 45 degrees with respect to the ground. The path

of the baseball is given by the function  

0f x 2.002 5 . What is the maximum

height reached by the ball?

x x

Real-Life Example:

Since this parabolic path is opening downward, the maximum height is reached

at the vertex point. You can use the formulas for h and k to find the vertex point. Then, the maximum height is represented by the k value.

Real-Life Example:

The maximum height reached by this ball

is 130 ft.

20.002 5f x x x

20.002 5f x x x

1

2502 0.002

h

2250 0.002 250 250 5 130 .k f ft

Graph:

The maximum height reached by this ball is

130 ft.

20.002 5f x x x

50 100 150 200 250 300 350 400 450 500 550-50

25

50

75

100

125

150

-25

-50

x

y

Real-Life Example:

2

A yo yo was dropped at a point 20 inches

above the ground with a velocity of 2 inch

per second with respect to the ground. 

The path of the yo yo is given by the function

4 20 .  Find the minimum heif x x x

ght

the yo yo will reach.

Real-Life Example:

Since this parabolic path is opening upward,

the minimum height is reached at the vertex

point. You can use the formulas for h and k

to find the vertex point. Then, the minimum

height is represented by the k value.

4

2 .2 1

h in

2 4 20f x x x

22 2 4 2 20 16 .k f in

Real-Life Example:

The minimum height reached by the yo-yo is

16 feet.

2 4 20f x x x

1 2 3 4 5 6 7 8 9-1-2-3-4-5

4

8

12

16

20

24

28

32

36

40

-4

x

y

Real-Life Example:

The minimum height reached by the yo-yo is

16 feet.

2 4 20f x x x

1 2 3 4 5-1

4

8

12

16

20

24

x

y