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.