The presentation includes all the necessary material for review of Chapter 4 Quadratics for A-level Pure Mathematics Cambridge International Examinations. It will be useful for students as the review materials and also for teachers as the teaching tips and presentations in the class.
The presentation was made by Oleksii Khlobystin (Alex) - Mathematics Teacher at Depu Foreign Language School, Chongqing City, China.
Personal website: www.visualcv.com/o-khlobystin
Personal email: o-khlobystin@yandex.comPresentation POWER POINT Chapter 4 Quadratics, A-level Pure Mathematics CIE 9709
Made by Oleksii Khlobystin (Alex) - Mathematics Teacher at Depu Foreign Language School – Cambridge International Centre, Chongqing City, China.
Personal website: www.visualcv.com/o-khlobystin
Personal email: o-khlobystin@yandex.com
The graph of forms a curve called a
parabola
y
2x
y
2x
This point . . . is called the vertex
e.g.
y
2x
32xy2xy2xyAdding a constant translates up the y-
values2xy32xy
The vertex is now ( 0,
has added 3 to the y-
axis
y
2x
2 x
y
3
3)
Adding 3 to x
gives
2x
y
We
get
23)(xy 2x
Adding 3
to x
moves the
curve 3 to
the left.
This may seem surprising but on the x-axis,
y = 0
so,
0y
y
( x
0
( x
23
)
23)
x
3
y
2x
y
Translating in both
y
(x
5 2
)
3
5 2
)
3
(x
y
2x
directions
y
e.g. 35We can write this in vector form
translation
as:
y qpq
2)
p
2x
qp
),
SUMMARY
(
x
The curve
y
is a translation of by
The vertex is given by
(
SUMMARY
Exercises: Sketch the following translations of
y
1.
y
2x
y
(x
2 2
)
1
2.
y
2x
y
(x
3 2
)
2
3.
y
2x
y
(x
4 2
)
3
2x
1)2(2xy 2xy2xy2)3(2xy2xy3)4(2xy
y 32212x
y
2x
y
(x
2 2
)
3
y
(x
1 2
)
2
4 Sketch the curve found by translating
by . What is its equation?
5 Sketch the curve found by translating
by . What is its equation?
y
p
q
(
x
2)
A quadratic function which is written in
the form
is said to be in its completed square
form.
We often multiply out the brackets as
follows:
This means
multiply ( x – 5 )
by itself
(x
5 2
)
e.g.
3
)
x
5
5
)(
x5x5
3
25
3
y
y
y
x
(
2x
2
x
y
10
x
28
The completed square form of a
quadratic function
• writes the equation so we can see the
translation from
y
• gives the vertex
2x
e.g. Consider translated by 2 to
y
2x
the left and 3 up.
Check: The vertex is ( 2, 3)
We can write this in
vector form as:
2
translation
3
The equation of the
curve is
y
3
Completed square form
2 2
)
(x
To write a quadratic function in
completed square form:
+7168422xxx)(But,
- 4 2
(
x
x
)
Half the coefficient of
x
So,
Subtract
16 to get
rid of (-4)2
Check by multiplying out!
7
- 8
x
8
x
(x
7
- 16
2
4
)
2
x
9
e.g.
2
To write a quadratic function in
completed square form:
SUMMARY
2
x
e.g.
bx
2
x
c
6
x
3
• Draw a pair of brackets
containing x with a
square outside.
• Insert the sign of b
and half the value
of b.
subtract it.
• Square half of b and
• Add c.
• Collect terms.
( x
2)
(
x
2)3
(
(
(
x
x
x
)3
)3
)3
2
2
2
9
39
6
SUMMARY
Exercise
s
1.
2.
6
x
4
Complete the square for the following
quadratics:
2
2
2
2
6
2
)
x
2 2
64
)
2 2
2
)
2
2
2
2
3
)
2 2
34
)
2 2
7
)
2
2
3
3
)
3 2
9
)
3 2
1
)
(x
(x
(x
(x
(x
(x
(x
(x
(x
x
6
10
2
x
2
x
3.
4
x
3
10
10
Exercises
4.
5.
6.
2
x
8
x
2
2
x
3
x
3
(x
(x
(x
(x
4 2
)
4 2
)
2
)
2
3
2
3
)
2
2
16
18
3
9
4
3
4
8
x
1
2 2
x
Hint: Start by taking 2 out as
though it were a common
factor
4 2
)
4 2
)
2
)
2
3
2
3
)
2
4.
5.
6.
2
x
8
x
2
2
x
3
x
3
2 2
x
8
x
1
(x
(x
2
(x
(x
x
2
(x
2
(x
2
(x
2
1
2
4
2
7
7
x
4
2
2
)
22
)
2
2
)
2
16
18
9
4
3
4
3
2
1
Chapter 4 Quadratics , Translations and Completing the Square-Alex.ppt
