Transcript Relating the Standard and Vertex Forms: Completing the Square

Forms, Forms and Forms

What forms of quadratic equations have
we learnt about so far?
Standard Form y=ax2+bx+c
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Factored Form y=a(x-r)(x-s)
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What useful info?
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Remember, Standard form tells us some
useful things:
1) Direction of opening
2) y intercept
What useful info?
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Remember, factored form also tells us
some useful info
1) Direction of opening
2) X intercepts
Now there is another form to
make your life more convenient
It is called Vertex Form!
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Here it is:
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y=a(x-h)2 + k
What does it all mean?

2
y=a(x-h)
Direction of
opening
+k
X coordinate of vertex.
Watch the sign
Y coordinate of vertex
Can you see why it might be
useful?
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Vertex form tells us
1) Direction of opening
2) Coordinates of the Vertex
Example
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Given y=-2(x-4)2 +5
What is the direction of opening?
Down
What are the coordinates of the vertex?
(4,5)
Why is it (+4)
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Because if the vertex is at (4,5) the (-)
sign in front of the "h" will make (+4)
show up as (-4)
One more example
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Given y=0.5(x+3)2 -8
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What are the coordinates of the vertex?
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(-3,-8)
Relating the Standard and Vertex
Forms: Completing the Square
Unfortunately…..there are some
things in life that you just have to
remember…..
Step 1
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Insert brackets around the first two terms
y = (x +12x)+ 40
2
Step 2
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Factor out any value in front of the x2
In this case there is nothing to factor out
y = (x +12x) + 40
2
Step 3
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Take Middle term divide by 2 & square it
12
=6
2
2
6 = 36
Step 4

Rewrite the equation with the result
from step 3 added and subtracted inside
the brackets
y = (x +12x + 36 - 36) + 40
2
Step 5

Bring the (-) term outside of the bracket
remember to remultiply if necessary
y = (x +12x + 36)- 36 + 40
2
Step 6
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Combine the two constant terms outside the
bracket
y = (x +12x + 36)+ 4
2
Step 7
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Factor trinomial inside the bracket
y = (x + 6)(x + 6) + 4
Step 8
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Express answer in vertex form
y = (x + 6) + 4
2
Step 1
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Insert brackets around the first two terms
Y =2x2 + 36x + 170
Y = (2x2 + 36x) + 170
Step 2
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Factor out any value in front of the x2
In this case there is something to factor
out
Y = 2(x2 + 18x) + 170
Step 3

Take Middle term divide by 2 & square it
18
=9
2
2
9 = 81
Step 4

Rewrite the equation with the result from
step 3 added and subtracted inside the
brackets
Y = 2(x2 + 18x + 81 – 81) + 170
Step 5

Bring the (-) term outside of the bracket
remember to multiply if necessary. This
time it is!!!!
Y = 2(x2 + 18x + 81) – 162 + 170
Step 6
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Combine the two constant terms outside
the bracket
Y = 2(x2 + 18x + 81) + 8
Step 7
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Factor trinomial inside the bracket
Y = 2 (x + 9) ( x + 9) + 8
Step 8
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Express answer in vertex form
Y = 2(x + 9)2 + 8