quadratic equations, inequalities and functions module 1 lesson 1 quadratic functions
TRANSCRIPT
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
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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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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)}
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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.
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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.
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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)
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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)
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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.
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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
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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