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Transcript of 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS...
![Page 1: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/1.jpg)
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
![Page 2: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/2.jpg)
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"
![Page 3: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/3.jpg)
Translations
2xy
The graph of forms a curve called a parabola
2xy
This point . . . is called the vertex
![Page 4: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/4.jpg)
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
![Page 5: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/5.jpg)
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
![Page 6: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/6.jpg)
Translations
Translating in both directions 35 2 )(xy2xy e.g.
3
5
We can write this in vector form as:
translation
35 2 )(xy2xy
![Page 7: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/7.jpg)
Translations
SUMMARY
The curve
is a translation of by 2xy
q
pqpxy 2)(
The vertex is given by ),( qp
![Page 8: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/8.jpg)
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
![Page 9: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/9.jpg)
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
![Page 10: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/10.jpg)
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
![Page 11: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/11.jpg)
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
![Page 12: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/12.jpg)
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
![Page 13: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/13.jpg)
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
![Page 14: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/14.jpg)
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
![Page 15: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/15.jpg)
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
![Page 16: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/16.jpg)
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
![Page 17: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/17.jpg)
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)
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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
![Page 19: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/19.jpg)
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
![Page 20: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/20.jpg)
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
![Page 21: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/21.jpg)
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:
![Page 22: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/22.jpg)
Completing the Square
282 xx
332 xx
182 2 xx
184 2 )(x
432
23 )(x
722 2 )(x
4.
5.
6.
![Page 23: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/23.jpg)
Completing the Square
![Page 24: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/24.jpg)
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.
![Page 25: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/25.jpg)
Translations and Completing the Square
SUMMARY
The curve
is a translation of by 2xy
q
pqpxy 2)(
The vertex is given by ),( qp
![Page 26: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/26.jpg)
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
![Page 27: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/27.jpg)
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
![Page 28: 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.](https://reader035.fdocuments.in/reader035/viewer/2022070411/56649f505503460f94c729e2/html5/thumbnails/28.jpg)
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.