1. Translations and Completing the Square © Christine Crisp

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

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

4. 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.

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

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

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

8. Translations Sketch the following translations of 1. Now insert a coefficient infront of the bracket The 2 outside the bracket has stretched the curve vertically by a factor of 2

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 So

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 If the graph is plotted we find that the vertex is at (–1, 3) The graph has a Horizontal translation of –1 so q = +1 Opposite sign Vertical translation of 3 so r = 3 Same Sign 2x 2 + 4x + 5 = p(x + q) 2 + r p = coefficient of x 2 p = 2 So 2x 2 + 4x + 5 = p(x + q) 2 + r = 2(x + 1) 2 + 3 = 2(x + 1) 2 + 3 We need to find the values of p, q and r

15. Completing the Square Using a Calculator to Complete the Square 1)Plot the given curve in y1 2) Use the window button to set the scale so that the vertex is clearly visible 3) Use the 2ndF Trace button (Calc) to find the vertex – either a maximum or a minimum 4) Fill in the horizontal translation q 5) Fill in the vertical translation r 6) Fill in the vertical stretch using the coefficient of x 2

16. Completing the Square Ex1 y = 2x 2 – 3x – 5 Express in the form p(x + q) 2 + r Using 2ndF Trace button (Calc) to find the vertex Min at x = 0.75 and y = –6.125 Horizontal translation of + 0.75 so q = –0.75 Opposite sign Vertical translation of –6.125 so r = –6.125 Same Sign p = coefficient of x 2 = 2 So 2x 2 – 3x – 5 = 2(x – 0.75) 2 – 6.125

17. Completing the Square Ex1 y = 4 – 3x – x 2 Express in the form p(x + q) 2 + r Using 2ndF Trace button (Calc) to find the vertex Max at x = –1.5 and y = 6.25 Horizontal translation of –1.5 so q = 1.5 Opposite sign Vertical translation of 6.25 so r = 6.25 Same Sign p = coefficient of x 2 = –1 So 4 – 3x – x 2 = –1(x + 1.5) 2 + 6.25

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

19. Completing the Square 4. 5. 6. Qu.s in notes pg 32