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
"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"
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.
Top Related