4: translations and completing the square © christine crisp “teach a level maths” vol. 1: as...
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4: Translations and 4: Translations and Completing the SquareCompleting the Square
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
Translations and Completing the Square
Module C1
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Translations
2xy
The graph of forms a curve called a parabola
2xy
This point . . . is called the vertex
Translations
32 xy2xy
2xy
Adding a constant translates up the y-axis
2xy 32 xye.g.
2xy
The vertex is now ( 0, 3)
has added 3 to the y-values
2xy 32 xy
Translations
This may seem surprising but on the x-axis, y = 0so, x 3
We get
230 )( x0y
Adding 3 to x gives 23)( xy2xy
Adding 3 to x moves the curve 3 to the left.
23)( xy
2xy
Translations
Translating in both directions 35 2 )(xy2xy e.g.
3
5
We can write this in vector form as:
translation
35 2 )(xy2xy
Translations
SUMMARY
The curve
is a translation of by 2xy
q
pqpxy 2)(
The vertex is given by ),( qp
Translations
Exercises: Sketch the following translations of 2xy
12 2 )(xy2xy 1.
23 2 )(xy2xy 2.
34 2 )(xy2xy 3.
1)2( 2 xy
2xy
2xy
2)3( 2 xy
2xy
3)4( 2 xy
Translations
4 Sketch the curve found by translating2xy
3
2
2
12xy
by . What is its equation?
5 Sketch the curve found by translating
by . What is its equation?
32 2 )(xy
21 2 )(xy
Translations and Completing the Square
We often multiply out the brackets as follows: 35 2 )(xye.g.
3
355 ))(( xxy
28102 xxy
y x5x5 252x
A quadratic function which is written in the form qpxy 2)(is said to be in its completed square form.
This means multiply ( x – 5 ) by itself
Completing the Square
The completed square form of a quadratic function
• writes the equation so we can see the translation from
2xy • gives the vertex
Completing the Square
e.g. Consider translated by 2 to the left and 3 up.
2xy
The equation of the curve is 32 2 )(xy
Check: The vertex is ( -2, 3)
3
2
We can write this in vector form as:
translation
Completed square form
Completing the Square
= 2(x2 + x + x + 1) + 3= 2(x2 + x + x
Any quadratic expression which has the form ax2 + bx + c can be written as p(x + q)2 + r
2x2 + 4x + 5 = 2(x + 1)2 + 3
This can be checked by multiplying out the bracket
2(x + 1)2 + 3 = 2(x + 1)(x + 1) + 3
= 2(x2
= 2x2 + 4x + 2 + 3
= 2x2 + 4x + 5
= 2(x2 + x
Completing the Square
We have to find the values of p, q and r
p(x + q)2 + r = p(x + q)(x + q) + r
= px2 + 2pqx + pq2 + r
= p(x2 + 2qx + q2) + r
Match up your expression with this one to findp , q and r
MethodExpand p(x + q)2 + r
Completing the Square
Express x2 + 4x + 7 in the form p(x + q)2 + r
Obviously p = 1 to obtain 1x2
x2 + 4x + 7 = p(x + q)2 + r
x2 + 4x + 7 = 1(x + q)2 + r
= x2 + 2qx + q2 + r
= 1(x + q)(x + q) + r
= x2= x2 + qx= x2 + qx + qx= x2 + qx + qx + q2= x2 + qx + qx + q2 + r
Completing the Square
x2 + 4x + 7 = p(x + q)2 + r = 1(x + 2)2 + 3
22 + r = 7
matching up the x terms
q = 2
matching up the number terms
r = 7 – 4 = 3
2qx = 4x
q2 + r = 7
subst. q = 2
x2 + 4x + 7 = x2 + 2qx + q2 + r
divide by 2x
To find the values of q and r match up the terms
Completing the Square
-1
1
2
3
4
5
6
7
8
9
-1-2-3-4 1 2 3 4 50
Graphing the resultant equation 1(x + 2)2 + 3
y = x2y = (x + 2)2y = (x + 2)2 + 3
Horizontal translation of -2Vertical translation of +3Vertex (-2, 3)
Completing the Square
Express 2x2 - 6x + 7 in the form p(x + q)2 + r
Obviously p = 2 to obtain 2x2
2x2 - 6x + 7 = 2(x + q)2 + r
= 2(x2 + 2qx + q2) + r
= 2(x + q)(x + q) + r
= 2x2 + 4qx + 2q2 + r
So 2x2 - 6x + 7 = 2x2 + 4qx + 2q2 + r
2x2 - 6x + 7 = p(x + q)2 + r
Completing the Square
matching up the x terms
matching up the number terms
4qx = -6x
2q2 + r = 7
subst. q = -
2x2 - 6x + 7 = p(x + q)2 + r = 2(x - )2 + 2
2(- )2 + r = 7
2x2 - 6x + 7 = 2x2 + 4qx + 2q2 + r
r = 9
227 -
divide by 4x
To find the values of q and r match up the terms
q = = -6
4
Completing the Square
-1
1
2
3
4
5
6
7
8
9
-1-2-3-4 1 2 3 4 50
Graphing the resultant equation 2(x - )2 + 2
y = x2
y = 2(x - )2 + 2
Horizontal translation +Vertical stretch factor 2Vertical translation +2 Vertex (, 2)
y = 2(x - )2
y = (x - )2
Completing the Square
642 xx
342 xx
1.
2.
3. 1062 xx
22 2 )(x
72 2 )(x
13 2 )(x
ExercisesComplete the square for the following quadratics:
Completing the Square
282 xx
332 xx
182 2 xx
184 2 )(x
432
23 )(x
722 2 )(x
4.
5.
6.
Completing the Square
Translations and Completing the Square
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Translations and Completing the Square
SUMMARY
The curve
is a translation of by 2xy
q
pqpxy 2)(
The vertex is given by ),( qp
Translations and Completing the Square Translating in both
directions 35 2 )(xy2xy e.g.
3
5
We can write this in vector form as:
translation
35 2 )(xy
2xy
Translations and Completing the Square SUMMARY
• Draw a pair of brackets containing x with a square outside.
• Insert the sign of b and half the value of b.
2)( x
2)3( x
• Square the value used and subtract it.
• Add c.
9)3( 2 x39)3( 2 x
• Collect terms. 6)3( 2 x
362 xxe.g.
To write a quadratic function in completed square form:
cbxx 2
Translations and Completing the Square SUMMARY
e.g.
342 xx342 2 )(x
72 2 )(x
322 22 )(x
Completing the Square
182 2 xx 212 42 xx
212 4)2(2 x
272)2(2 x
722 2 )(x
e.g.