Quadratic functions - Nguyễn Thị Thái Phượng - Hà Nội - Việt Nam

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Teacher

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Do you know graph of Quadratic Functions?
y
xO
y
x
O
y = ax2
( a > 0) y = ax2
( a < 0 )
Whether the graph opens up or down?
Recall back

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xO
y
a > 0
x
O
y
a < 0
Parabola
2
y ax=
. Vertex O(0;0)
.Axis of symmetry: x=0
.Parabola opening up when a>0,
opening down when a < 0

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GRAPH OF QUADRATIC
FUNCTIONS
Graph of quadratic function is a parabola
Example: Parabola

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PARABOLPARABOL

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ARCH GATE
( USA )

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PARABOLPARABOL

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PARABOLAPARABOLA

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PARABOLAPARABOLA

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PARABOLAPARABOLA

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PARABOLAPARABOLA

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PARABOLAPARABOLA

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PARABOLAPARABOLA

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Graph of Quadratic FunctionsI
1. Comment
2.Graphs
Variation trend of Quadratic FunctionsII
3. How to
graph
QUADRATIC FUNCTIONS

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a.Definition:
Function f(x) has the form ax2
+ bx + c
(a ≠ 0) a,b and c are real numbers,
then f is called a quadratic function.
f(x)= ax2
+ bx + c (a ≠ 0)
Domain:D= R
I.GRAPH OF
QUADRATIC
FUNCTIONS
a. Definition
QUADRATIC FUNCTIONS

16. ExampleExample
• f(x)=2x2
-5x + 3
• g(x)=x2
-2x
• h(x)=x2
-2x
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xO
y
a > 0
x
O
y
a < 0
b) Review .Parabola
2
y ax=
. Vertex O(0;0)
.Axis of symmetry: x=0
+ Parabola opening up when a > 0
opening down when
a < 0
QUADRATIC FUNCTIONS
I. Graph of
quadratic
functions

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x
O
y
xO
y
2
b
a
−
4a
∆
−
4a
∆
−
2
b
a
−
Graph of Quadratic Functions
y = ax2
+ bx +c (a ≠ 0 )
a > 0 a < 0
I
I
Find the coordinates of the vertex of
parabola?
QUADRATIC FUNCTIONS
I.QUADRATIC
FUNCTIONS
a. Definition
1. Comments
b. Review
2
y ax bx c= + +
Parabola
Vertex O(0;0)
Symmetric axis: x=0
Parabola opening up
when a > 0 opening
down when a < 0
2
y ax=
(a ≠ 0 )

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c) Vertex of Parabola
Vertex
Of graph of function
;
2 4
b
I
a a
∆ 
− − ÷
 
2
y ax bx c= + +
QUADRATIC FUNCTIONS
I.Graph of
quadratic
functions
a. Definition
1.
Comment
b. Review
Parabola
Vertex O(0;0)
Symmetric axis : x=0
Parabola opening
up when a > 0
opening down when
a < 0
2
y ax=
Note: ( )
4 2
b
y
a a
∆
− = −
2
y ax bx c= + +
(a ≠ 0 )
:
(a ≠ 0 )
:

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xO
y
a > 0
xO
y
a < 0
2
b
a
−
4a
−∆
I
2
b
x
a
−
=
2
b
a
−
4a
−∆ I
2
b
x
a
−
=
.
.
2.Graph
Đồ thị hàm số
2
y ax bx c= + +
The graph of the function
(a ≠ 0 ) is a parabola with the vertex being at point
, and the axis of symmetry being a line
.
. This parabola is an open up when a>0, open
down when a<0.
;
2 4
b
I
a a
∆ 
− − ÷
 
2
b
x
a
= −
QUADRATIC FUNCTIONS
I. Graph of
quadratic
functions
2
y ax bx c= + +

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a) Vertex Axis of symmetry
x = 3
2
) 6 5a y x x= − +
2
) 6 5b y x x= − + −
QUADRATIC FUNCTIONS
I. Graph of
quadratic
functions
Graph
.Vertex ;
2 4
b
I
a a
∆ 
− − ÷
 
. Symmetric axis:
2
b
x
a
= −
. Parabola opening
up a > 0,
Opening down when
a < 0
Ans
Ex 1: Identify vertex and axis of
symmetry of graphs
(3; 4);I −
b) Vertex Axis of symmetry
x = 3
(3;4);I

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Ex 2: Choose the correct option in the following
exercise:
Graph of function has the symmetry axis
is a line
2
2 3 1y x x= + +
(A).
3
2
x =
(B).
3
4
x = −
(C).
(D).
3
4
x =
3
2
x = −
Key: B

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x
O
y
xO
y
2
b
a
−
4a
∆
−
4a
∆
−
2
b
a
−
Graph of function y = ax2
+ bx +c(a ≠
0)
a > 0 a < 0
I
I
How to graph quadratic functions y =
ax2
+ bx + c ?
2
b
a
−
4a
∆
−
I
QUADRATIC FUNCTIONS
I. Graph of
quadratic
functions
Graph
.Vertex ;
2 4
b
I
a a
∆ 
− − ÷
 
. Axis of symmetry:
2
b
x
a
= −
+ Parabol open up
when a > 0
Open down when a
< 0

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To graph the parabola y = ax2
+ bx +c (a≠0), we
take the following steps:
3. Determine the intersection of parabola with the y-axis
(0;c)) and with the x-axis (if any).
Determine some more points on the graph such as the
point of symmetry (0;c) through the axis of symmetry of
the parabola to sketch a graph more exactly.
QUADRATIC FUNCTIONS
1.Determine the coordinates of vertex ;
2 4
b
I
a a
∆ 
− − ÷
 
2.Sketch the axis of symmetry
2
b
x
a
= −
I. Graph of
quadratic
functions
Graph
.Vertex ;
2 4
b
I
a a
∆ 
− − ÷
 
.Axis of symmetry
2
b
x
a
= −
. Parabola opening
up when a > 0
Opening down when
a < 0
3 How to graph quadratic functions :
4. Sketch the parabola
Determine the concavity direction
of parabola

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Sketch the parabola :
EX 1: Sketch of function y = x2
– 4x + 3 (p)
Solution:
1
4a
∆
− = −1) 2;
2
b
B
a
− = Vertex I( 2 ; -
1)
2)Axis of symmetry : x = 2
3) Intersections of (p)
with the x-axis : (1;0);
(3;0)
-y-axis : (0;3)
-The point of symmetry (0;3)
through the axis of symmetry
( 4;3)
3
3-1
2
4
O
x
y
I
QUADRATIC FUNCTIONS
I. Graph of
quadratic
function
Sketch:
1.Vertex
;
2 4
b
I
a a
∆ 
− − ÷
 
2.Axis of symmetry
2
b
x
a
= −
3. Intersections of
parabola with the y-
axis (point (0;c)) and
with the x-axis(if
any).
4. Sketch the
parabola

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Group 1: Determine the
coordinates of vertex and
its y-intercept and x-
intercept (if any) of
parabola
y = x2
– 3x + 3
y = x2
– 3x + 3
Group 2: Determine the
coordinates of vertex and
its y-intercept and x-
intercept (if any) of
parabola
y = x2
– 2x
Group 3: Determine the
coordinates of the vertex and
function of the axis of
symmetry of parabola
Group 4: Determine the
coordinates of intersection of
parabola
a) With the y-axis ; b) With
the x-axis
2
4 3y x x= − + −2
4 3y x x= − + −

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Wrap upWrap up
I.Graph of quadratic
functions
a. Definition:
1. Comment
Parabola
Vertex O(0;0)
Axis of symmetry: x=0
Parabola opening up when a
> 0 opening down when a < 0
2
y ax=
c)
2
y ax bx c= + +
b. Recall back
2.Graph
. Vertex
.Axis of symmetry:
2
b
x
a
= −
+ Parabol opening
up when a > 0
Opening down when
a < 0
;
2 4
b
I
a a
∆ 
− − ÷
 
3 How to graph:
1.Determine vertex
;
2 4
b
I
a a
∆ 
− − ÷
 
2.Sketch axis of
symmetry
2
b
x
a
= −
3. Determine the
intersections of
parabola with the y-
axis (point (0;c)) and
x-axis (if any).
4. Sketch parabola

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Home work
Exercises 1, 3 Text book page 49

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M ƠN QUÍ THẦY CÔ VÀ CÁC EM ĐÃ THAM GIA BÀI HỌM ƠN QUÍ THẦY CÔ VÀ CÁC EM ĐÃ THAM GIA BÀI HỌM ƠN QUÍ THẦY CÔ VÀ CÁC EM ĐÃ THAM GIA BÀI HỌThank you so much

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Group 1: Determine the coordinates of vertex and its
y-intercept and x-intercept (if any) of parabola
y = x2
– 3x + 3
Ans:Vertex
The y-intercept ( 0;3)
Parabola not cut x-axis
3 3
;
2 4
I
 
 ÷
 

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Group 2: Determine the coordinates of vertex and
its y-intercept and x-intercept (if any) of parabola
y = x2
– 2x
Ans: Vertex I(1;-1)
The y-intercept is Oy ( 0;0)
The x-intercept are (0;0) ; (2;0)

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Group 3: Determine the coordinates of the vertex and function
of the axis of symmetry of parabola
2
4 3y x x= − + −
Ans: Vertex I(2; 1)
Symmetric axis is a line x = 2

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Group 4: Determine the coordinates of intersection of
parabola
a) With the y-axis ; b) With the x-axis
2
4 3y x x= − + −
Ans:
With the y-axis ( 0;-3)
With the x-axis (1;0) ; (3;0)