1. 4: Translations and Completing the Square © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

2. 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"

3. Translations The graph of forms a curve called a parabola This point...is called the vertex

4. Translations  Adding a constant translates up the y -axis e.g. The vertex is now ( 0, 3) has added 3 to the y -values

5. Translations This may seem surprising but on the x -axis, y = 0 so, We get Adding 3 to x gives Adding 3 to x moves the curve 3 to the left.

6. Translations  Translating in both directions e.g. We can write this in vector form as: translation

7. Translations SUMMARY  The curve is a translation of by  The vertex is given by

8. Translations Exercises: Sketch the following translations of 1.2.3.

9. Translations 4 Sketch the curve found by translating by. What is its equation? 5 Sketch the curve found by translating by. What is its equation?

10. Translations and Completing the Square We often multiply out the brackets as follows: e.g. A quadratic function which is written in the form is said to be in its completed square form. This means multiply ( x – 5 ) by itself

11. Completing the Square The completed square form of a quadratic function writes the equation so we can see the translation from gives the vertex

12. Completing the Square e.g.Consider translated by 2 to the left and 3 up. The equation of the curve is Check: The vertex is ( -2, 3) We can write this in vector form as: translation Completed square form

13. Completing the Square = 2(x 2 + x + x + 1) + 3= 2(x 2 + x + x Any quadratic expression which has the form ax 2 + bx + c can be written as p(x + q) 2 + r 2x 2 + 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(x 2 = 2x 2 + 4x + 2 + 3 = 2x 2 + 4x + 5 = 2(x 2 + x

14. 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 = px 2 + 2pqx + pq 2 + r = p(x 2 + 2qx + q 2 ) + r Match up your expression with this one to find p, q and r Method Expand p(x + q) 2 + r

15. Completing the Square Express x 2 + 4x + 7 in the form p(x + q) 2 + r Obviously p = 1 to obtain 1 x 2 x 2 + 4x + 7 = p(x + q) 2 + r x 2 + 4x + 7 = 1(x + q) 2 + r = x 2 + 2qx + q 2 + r = 1(x + q)(x + q) + r = x 2 = x 2 + qx = x 2 + qx + qx = x 2 + qx + qx + q 2 = x 2 + qx + qx + q 2 + r

16. Completing the Square x 2 + 4x + 7 = p(x + q) 2 + r = 1(x + 2) 2 + 3 2 2 + r = 7 matching up the x terms q = 2 matching up the number terms r = 7 – 4 = 3 2qx = 4x q 2 + r = 7 subst. q = 2 x 2 + 4x + 7 = x 2 + 2qx + q 2 + r divide by 2x To find the values of q and r match up the terms

17. Completing the Square Graphing the resultant equation 1(x + 2) 2 + 3 y = x 2 y = (x + 2) 2 y = (x + 2) 2 + 3 Horizontal translation of -2 Vertical translation of +3 Vertex (-2, 3)

18. Completing the Square Express 2x 2 - 6x + 7 in the form p(x + q) 2 + r Obviously p = 2 to obtain 2x 2 2x 2 - 6x + 7 = 2(x + q) 2 + r = 2(x 2 + 2qx + q 2 ) + r = 2(x + q)(x + q) + r = 2x 2 + 4qx + 2q 2 + r So 2x 2 - 6x + 7 = 2x 2 + 4qx + 2q 2 + r 2x 2 - 6x + 7 = p(x + q) 2 + r

19. Completing the Square matching up the x terms matching up the number terms 4qx = -6x 2q 2 + r = 7 subst. q = -  2x 2 - 6x + 7 = p(x + q) 2 + r = 2(x -  ) 2 + 2  2(-  ) 2 + r = 7 2x 2 - 6x + 7 = 2x 2 + 4qx + 2q 2 + r r = 7 -  divide by 4x To find the values of q and r match up the terms q = = - 

20. Completing the Square Graphing the resultant equation 2(x -  ) 2 + 2  y = x 2 y = 2(x -  ) 2 + 2  Horizontal translation +  Vertical stretch factor 2 Vertical translation +2  Vertex ( , 2  ) y = 2(x -  ) 2 y = (x -  ) 2

21. Completing the Square 1. 2. 3. Exercises Complete the square for the following quadratics:

22. Completing the Square 4. 5. 6.

23. Completing the Square

24. 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.

25. Translations and Completing the Square SUMMARY  The curve is a translation of by  The vertex is given by

26. Translations and Completing the Square  Translating in both directions e.g. We can write this in vector form as: translation

27. 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. Square the value used and subtract it. Add c. Collect terms. e.g.  To write a quadratic function in completed square form:

28. Translations and Completing the Square SUMMARY e.g.  Completing the Square e.g.