quadratic equations, inequalities and functions module 1 lesson 1 quadratic functions

Quadratic

Equations,

Inequalities and

Functions

Module 1

Lesson 1

Quadratic Functions

2

LAS # 2: STANDARD Form of a Quadratic Function

The function defined by the second degree equation f(x) = ax2 + bx + c

where a, b, and c are real numbers and a ≠ 0, is a quadratic function in x.

This function can also be written as y = ax2 + bx + c , where y = f(x).

Example:

1. y = 2x2 – 3x - 10

2. y = 3x2 + 5x

3. y = 4x2 - 7

5. y = x2

a b c

2 -3 -10

3 5 0

4 0 -7

-5 0

1 0 0

4. y = -5x2 – 3x 2 2

-32

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3

STANDARD FORM of a Quadratic Function

Example: Write the following equations in STANDARD FORM.

1. y – 12 = - 5x + 3x2

y = – 5x + 3x2 + 12

2. y = (x + 5)2 – 2(x + 17) + 55

y = x2 + 10x + 25

y = x2 + 8x + 46

3. y = 7 – (x – 3) (x + 3)

y = 7 – (x2 – 9)

y = 7 – x2 + 9

y = -x2 + 16

f (x) = 3x2 – 5x + 12

(a + b)2 = a2 + 2ab + b2

(x + 5)2 = x2 + 2(x)(5) + 52

(x + 5)2 = x2 + 10x + 25

– 2(x + 17) = -2x - 34

y = 3x2 – 5x + 12

(a + b) ( a – b) = a2 - b2

(x – 3) (x + 3)= x2 - 9

f(x) = x2 + 8x + 46

or f(x) = -x2 + 16

– 2x – 34 + 55

f(x) = ax2 + bx + c

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4

4. y = x – 2(3x – 1)2 - 5 (a – b)2 = a2 – 2ab +b2

(3x – 1)2 = (3x)2 – 2(3x)(1) +12

(3x – 1)2 = 9x2 – 6x +1

y = x – 2(9x2 – 6x +1) - 5

y = x – 18x2 + 12x -2 - 5

y = – 18x2 + 13x -7

f(x) = – 18x2 + 13x -7

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5

Equal Differences Method

x -2 -1 0 1 2

y 4 1 0 1 4

1 1 1 1

-3 -1 1 3

222 2nd Difference

1st Difference

x -6 -4 -2 0 2 4 6f(x ) -7 -3 1 5 9 13 17

222222

4444441st Difference

The ordered pairs represents a

QUADRATIC FUNCTION

1) (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)

Direction: Using the equal

differences method, determine which

of the following ordered pairs

represent a quadratic function.

2) (-6, -7), (-4, -3), (-2, 1), (0, 5), (2, 9), (4, 13), (6, 17)

3) (-4, -73), (-2, -13), (0, -1), (2, 11), (4, 71)

x -4 -2 0 2 4f(x ) -73 -13 -1 11 71

2222

60121260

48-48 0

4848

1st Difference

2nd Difference

3rd Difference

NOT A QUADRATIC FUNCTION – LINEAR FUNCTION

NOT A QUADRATIC FUNCTION – CUBIC FUNCTION

Copyright © by Mr. Florben G. Mendoza

6

LAS # 2

Activity # 1

1. y – 5 = - 2x + 4x2

2. y - 6 = (x + 3)2 – 2(x + 12)

3. y = 9 – (x – 4) (x + 4)

4. y + 5 = x – 3(2x – 1)2

5. y = 5 + x(x - 3) + (x – 5) 2

I. Write each of these quadratic function in general form, then identify

the real numbers a, b, and c.

General Form a b c

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7

II. Using the equal differences method, name if the given ordered pairs represents

the following:

(a) Linear Function (c) Cubic Function (e) Quintic Function

(b) Quadratic Function (d) Quartic Function

5. {(-3, 83), (-2, 18), (-1, 3), (0, 2), (1, 3), (2, 18), (3, 83)}

2. {(-5, 60), (-2, 15), (1, 6), (4, 33), (7, 96), (10, 195)}

3. {(-3, -243), (-2, -32), (-1, -1), (0, 0), (1, 1), (2, 32), (3, 243)}

4. {(-3, -49), (-2, -19), (-1, -5), (0, -1), (1, -1), (2, 1), (3, 11)}

1. {(-6, -23), (-4, -17), (-2, -11), (0, -5), (2, 1), (4, 7), (6, 13)}

Copyright © by Mr. Florben G. Mendoza

8

LAS # 3: Graphs of Quadratic Functions

The graph of a quadratic function offers an interpretation of the nature of its

zeros, its symmetry and other characteristics. The usual method of graphing by

plotting points can be used for this purpose.

Consider the following functions and their graph.

1. f(x) = x2 or y = x2

x

y

-2 -1 0 1 2

4 1 0 1 4 Parabola

Vertex: The axis of symmetry

intersects the parabola at a point

called the vertex. (highest or lowest

point of the graph)

Axis of Symmetry: The axis of

symmetry is the line that divide the

graph into two halves

Axis of Symmetry: x = 0

Vertex: (0, 0)

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Symmetric Points:

Points that has equal

distance from the axis

of symmetry.

9

2. f(x) = -x2 or y = -x2

x

y

-2 -1 0 1 2

-4 -1 0 -1 -4

1. f(x) = x2 or y = x2

x

y

-2 -1 0 1 2

4 1 0 1 4

Graphs of Quadratic Functions: (y = ax2)

• When a is positive, the graph opens upward.

• When a is negative, the graph opens downward.

• The graph of f(x)= x2 and f(x) = -x2 are reflections of each other

about the x-axis.

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10

1. f(x) = x2

x

y

-2 -1 0 1 2

4 1 0 1 4

Graphs of Quadratic Functions: (y = ax2)

2. f(x) =12

x2

x

y

-2 -1 0 1 2

2 0 412

12

3. f(x) = 2x2

x

y

-2 -1 0 1 2

8 2 0 2 8

As |a| increases the graph becomes narrower and closer to the y-axis.

Copyright © by Mr. Florben G. Mendoza

11

Given Vertex Axis of Symmetry

Direction of opening

Symmetric Points

Illustration(Graph)

1. y = 2x2

2. y = ½ x2

3. y = -3x2

(0, 0) x = 0 Upward(1, 2)

(-1, 2)

(0, 0) x = 0 Upward(1, ½ )

(-1, ½ )

(0, 0) x = 0 Downward(1, -3)

(-1, -3)

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y = ax2Vertex: (0,0) Axis of Symmetry: x = 0

12

Graphs of Quadratic Functions: y = x2 + k

y = x2 y = x2 - 2 y = x2 + 3

In general, if f(x) = x2 + k, the vertex

of the parabola lies on the y-axis, but

shifted vertically |k| units upward if

k > 0 and downward if k < 0.

Observe that the graph have

different vertices though all of them

are located along the y-axis.

1. y = x2 + 4

2. y = x2 - 5

3. y = x2 + 1

Vertex

(0, 4)

(0, -5)

(0, 1)

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y = x2 + k

13

Given Vertex Axis of Symmetry

Direction of opening

Symmetric Points

Illustration(Graph)

1. y = x2 + 1

2. y = x2 - 3

3. y = -x2 - 2

(0, 1) x = 0 Upward(1, 2)

(-1, 2)

(0, -3) x = 0 Upward

(-1, -2 )

(1, -2)

(0, -2) x = 0 Downward (1, -3)

(-1, -3)

Copyright © by Mr. Florben G. Mendoza

y = x2 + k Axis of Symmetry: x = 0 Vertex: (0,k)

14

Graphs of Quadratic Functions: y = (x - h)2

y = x2 y =(x - 3)2 y =(x + 2)2

Notice that the only difference in the

graphs is the position of the vertex of each

parabola. All vertices are on the x-axis,

but translated horizontally.

In general, if f(x) = (x - h)2, the vertex of

the parabola is at (h, 0)

3. y = (x + 5)2

Vertex

2. y = (x - 3)2

1. y = (x - 4)2

(3, 0)

(-5, 0)

(4, 0)

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y = (x-h) 2

15

Given Vertex Axis of Symmetry

Direction of opening

Symmetric Points

Illustration(Graph)

1. y = (x + 1)2

2. y = -(x – 3)2

3. y = (x + 2)2

(-1, 0) x = -1 Upward(0, 1)

(-2, 1)

(3, 0) x = 3 Downward

(2, -1 )

(4, -1)

(-2, 0) x = -2 Upward (-1, 1)

(-3, 1)

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y = (x - h)2Vertex: (h,0) Axis of Symmetry: x = h

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Given Vertex Axis of Symmetry

Direction of opening

Symmetric Points

Illustration(Graph)

1. y = x2 + 8x + 16

Check:

16 = 4 x 2 = 8

y = (x + 4)2

(-4, 0) x = -4 Upward (-3, 1)

(-5, 1)

17

Graphs of Quadratic Functions

y = (x – 3)2 + 1

x

y

0 1 2 3 4

10 5 2 1 2

5

5

6

10

Now consider the graph of

f(x) = a (x-h)2 + k, a ≠ 0.

Observe that the vertex is at (h, k) and

the axis of symmetry is at x = h.

Copyright © by Mr. Florben G. Mendoza

y = a(x-h)2 + k

Vertex Form

18

1. y = x2 – 4x + 5

Solution:

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Graphs of Quadratic Functions: y = a(x-h)2 + k

y = ( x2 – 4x) + 5

y = [( x2 – 4x + 4) – 4] + 5

y = [( x – 2)2 – 4] + 5

y = ( x – 2)2 – 4 + 5

Vertex:

(2, 1)

Axis of Symmetry:

Direction:

x = 2

upward

Symmetric Points: (1, 2)

(3, 2)

y = ( x – 2)2 + 1

19

Graphs of Quadratic Functions: y = a(x-h)2 + k

2. y = 2x2 – 4x - 5

Solution:

y = 2 ( x2 – 2x) - 5

y = 2 [( x2 – 2x + 1) – 1] - 5

y = 2 [( x – 1)2 – 1] - 5

y = 2 ( x – 1)2 – 2 - 5

y = 2 [( x – 1)2 – 1] - 5

y = 2 ( x – 1)2 – 7

Vertex:

(1, -7)

Axis of Symmetry:

Direction:

x = 1

upward

Symmetric Points: (0, -5)

(2, -5)

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20

3. y = -x2 - 2x - 3

Solution:

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y = - (x2 + 2x) - 3

y = - [(x + 1)2 – 1] - 3

y = - ( x + 1)2 -2

Vertex:

(-1, -2)

Axis of Symmetry:

Direction:

y = - (x + 1)2 + 1 - 3

(0, -3)

(-2, -3)

x = -1

Symmetric Points:

downward

y = - [(x2 + 2x +1) – 1]- 3

Graphs of Quadratic Functions: y = a(x-h)2 + k

21

Graphs of Quadratic Functions: y = a(x-h)2 + k

4. y = - 3x2 + 18x - 23

Solution:

y = - 3(x2 – 6x) - 23

y = - 3[(x2 – 6x +9) – 9]- 23

y = -3[(x – 3)2 – 9] - 23

y = - 3( x – 3)2 + 4

Vertex:

(3, 4)

Axis of Symmetry:

Direction:

y = - 3(x – 3)2 + 27 - 23

(2, 1)

(4, 1)

x = 3

Symmetric Points:

downward

Copyright © by Mr. Florben G. Mendoza

22Copyright © by Mr. Florben G. Mendoza

LAS # 4: Activity # 2

I. Complete the table below.Given Vertex Axis of

SymmetryDirection of opening

Symmetric Points

Illustration(Graph)

1. y = 2x2

2. y = -3x2 - 1

3. y = 5x2 + 3

4. y = (x – 7)2

3. y = 2(x + 5)2

4. y = x2 – 6x + 9

5. y = x2 - 10x +25

6. y = -x2 - 8x -16

9. y = (x – 1)2 + 5

10. y = -2(x + 2)2 - 3

23Copyright © by Mr. Florben G. Mendoza

LAS #5: Activity # 3

II. Complete the table below.

Given Vertex Form

Vertex Axis of Symmetry

Direction of opening

Symmetric Points

Illustration(Graph)

1. y = x2 – 6x - 1

2. y = x2 – 8x + 15

3. y = x2 + 10x + 20

4. y = 3x2 + 18x + 25

5. y = -2x2 + 8x - 5