Quadratic Functions3

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Quadratic Functions(3)

• What is a perfect square.
• How to make and complete the square.
• Sketching using completed square

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A perfect square
What do we get if we factorise x2 10x 25
This is called a perfect square because it can
be written as (x5)2.
Can you think of an expression for a perfect
cube??
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Solving Quadratic Equations

• We will now look at solving quadratic equations
  using completing the square method.

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Complete the square for y x2 10x 12
Use (x 5)2 x2 10x 25
x2 10x 12 x2 10x 25 - 13
x2 10x 12 (x 5)2 - 13
y (x 5)2 - 13
is complete square form
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Solving Equations using the completed square
Solve x2 10x 12 0
Complete the square ..
(x 5)2 - 13 0
(x 5)2 13
(x 5) ??13
x -5 ??13
x -5 ?13 or -5 - ?13
x -1.39 or -8.61
The solutions
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Complete the square for y x2 - 20x - 30
Use (x - 10)2 x2 - 20x 100
x2 - 20x - 30 x2 - 20x 100 - 130
x2 - 20x - 30 (x - 10)2 - 130
y (x - 10)2 - 130
is completed square form
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Activity 1
Mathematics Standard Level Page 29-30 Exercise
1h Q 1-2
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Complete the square for y 2x2 - 14x - 33
Adjust to make a single x2 y 2(x2 - 7x -
16.5)
Use (x - 3.5)2 x2 - 7x 12.25
x2 - 7x - 16.5 x2 - 7x 12.25 - 28.75
2(x2 - 7x - 16.5) 2((x - 3.5)2 - 28.75)
y 2((x - 3.5)2 - 28.75)
y 2(x - 3.5)2 57.5
is complete square form
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Solving Equations using the completed square
Solve 2x2 - 14x - 33 0
x2 - 7x 16.5 0 (divide both sides
by 2)
Complete the square (from previous slide)..
(x - 3.5)2 - 28.75 0
(x - 3.5)2 28.75
(x - 3.5) ??28.75
x 3.5 ??28.75
x 3.5 ?28.75 or 3.5 - ?28.75
x 8.86 or -1.86
The solutions
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Activity 1
Mathematics Standard Level Page 29-30 Exercise
1h Q 1-4
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Quadratic graphs
Investigate what happens when you change a and
b.
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Quadratic Graphs
Investigate what happens when you change the
value of k.
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Quadratic graphs
This is a translation of the graph ykx2 by the
vector
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Finding critical values on graphs

• Find the y-intercept
• Find the x-intercept(s)
• Find the vertex

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Finding the y-intercept
Intercepts y-axis when x0
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Finding the x-intercept(s)
Intercepts x-axis when y0
Does it factorise??
x-2 and x-8
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Finding the vertex
Find translation from yx2 by writing in
completed square form.
Vertex must be at (-5,-9)
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Finding critical values on graphs

• Find the y-intercept (0,16)
• Find the x-intercept(s) (-2,0) (-8,0)
• Find the vertex (-5,-9)

Now sketch this graph
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Sketching the graph