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.

35

We can write this in vector form as:

translation

35 2 )(xy2xy

TranslationsSUMMARY

The curve

is a translation of by 2xy

qp

qpxy 2)(

The vertex is given by ),( qp

TranslationsExercises: 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

32

212xy

by . What is its equation?

5 Sketch the curve found by translatingby . 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)

32

We can write this in vector form as:

translation

Completed square form

Completing the Square

e.g.

To write a quadratic function in completed square form:

Half the coefficient of x

- 4 22 )( xxx - 8 7 - 16

Subtract 16 to get rid of (-4)2

Check by multiplying out!

+7

1684 22 xxx )(But,9478 22 )(xxxSo,

Completing the Square

• Draw a pair of brackets containing x with a square outside.

• Insert the sign of b and half the value of b.

2)( x

SUMMARY

2)3( x• Square half of b 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

Completing the Square

642 xx

342 xx

1.

2.

3. 1062 xx

622 22 )(x

1093 2 )(x

342 2 )(x

22 2 )(x

72 2 )(x

13 2 )(x

642 2 )(x

322 22 )(x

1033 22 )(x

ExercisesComplete the square for the following quadratics:

Completing the Square

282 xx

332 xx

182 2 xx

3492

23 )(x

2164 2 )(x184 2 )(x

432

23 )(x

4.

5.

6.Hint: Start by taking 2 out as though it were a common factor

Completing the Square

282 xx

332 xx

182 2 xx

3492

23 )(x

212 42 xx

2164 2 )(x184 2 )(x

212 422 )(x

432

23 )(x

27222 )(x

722 2 )(x

4.

5.

6.

Completing the Square

Translations and Completing the Square

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

Translations and Completing the Square

SUMMARY The curve

is a translation of by 2xy

qp

qpxy 2)(

The vertex is given by ),( qp

Translations and Completing the Square Translating in both

directions 35 2 )(xy2xy e.g.

35

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.