Presentation POWER POINT Chapter 4 Quadratics, A-level Pure Mathematics CIE 9709

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  • 17.05.2018
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The presentation includes all the necessary material for review of Chapter 4 Quadratics for A-level Pure Mathematics Cambridge International Examinations. It will be useful for students as the review materials and also for teachers as the teaching tips and presentations in the class. The presentation was made by Oleksii Khlobystin (Alex) - Mathematics Teacher at Depu Foreign Language School, Chongqing City, China. Personal website: www.visualcv.com/o-khlobystin Personal email: o-khlobystin@yandex.comPresentation POWER POINT Chapter 4 Quadratics, A-level Pure Mathematics CIE 9709 Made by Oleksii Khlobystin (Alex) - Mathematics Teacher at Depu Foreign Language School – Cambridge International Centre, Chongqing City, China. Personal website: www.visualcv.com/o-khlobystin Personal email: o-khlobystin@yandex.com
Иконка файла материала Chapter 4 Quadratics , Translations and Completing the Square-Alex.ppt
The graph of forms a curve called a parabola y  2x y  2x This point . . . is called the vertex
e.g. y  2x 32xy2xy2xyAdding a constant translates up the y- values2xy32xy The vertex is now ( 0,  has added 3 to the y-  axis  y  2x 2 x y 3 3)
Adding 3 to x   gives 2x y  We get 23)(xy 2x Adding 3 to x moves the curve 3 to the left. This may seem surprising but on the x-axis, y = 0 so, 0y y (  x 0 (  x 23 ) 23)  x  3  y 
2x  y  Translating in both y  (x 5 2  ) 3 5 2  ) 3  (x y  2x directions y  e.g. 35We can write this in vector form translation     as:
y  qpq 2) p 2x qp ), SUMMARY ( x  The curve   y is a translation of by  The vertex is given by ( SUMMARY
Exercises: Sketch the following translations of y  1. y   2x y  (x 2 2  ) 1 2. y   2x y  (x 3 2  ) 2 3. y   2x y  (x 4 2  ) 3 2x 1)2(2xy 2xy2xy2)3(2xy2xy3)4(2xy
y  32212x y  2x y  (x 2 2  ) 3 y  (x 1 2  ) 2 4 Sketch the curve found by translating by . What is its equation? 5 Sketch the curve found by translating by . What is its equation?
y p q ( x  2)  A quadratic function which is written in the form is said to be in its completed square form. We often multiply out the brackets as follows: This means multiply ( x – 5 )  by itself  (x 5 2  ) e.g. 3 )  x 5 5 )( x5x5  3 25 3 y y y  x ( 2x 2  x   y 10 x  28
The completed square form of a quadratic function • writes the equation so we can see the translation from y  • gives the vertex 2x
e.g. Consider translated by 2 to y  2x the left and 3 up. Check:  The vertex is  ( ­2, 3) We can write this in vector form as:  2 translation      3     The equation of the curve is y 3 Completed square form  2 2  ) (x
To write a quadratic function in completed square form: +7168422xxx)(But, -  4 2 (         x x ) Half the coefficient of  x So, Subtract 16 to get rid of (-4)2 Check by multiplying out!            7  -  8        x 8 x  (x 7 - 16 2 4 ) 2 x   9 e.g. 2
 To write a quadratic function in  completed square form: SUMMARY 2 x e.g. bx 2 x  c 6 x  3 • Draw a pair of brackets containing x with a square outside. • Insert the sign of b and half the value of b. subtract it. • Square half of b and • Add c. • Collect terms. ( x 2) ( x 2)3 ( ( ( x x x )3 )3 )3 2  2 2   9 39 6 SUMMARY
Exercise s 1. 2.  6  x 4 Complete the square for the following quadratics: 2 2   2 2 6 2 ) x 2 2  64 ) 2 2  2 ) 2 2  2 2 3 ) 2 2  34 ) 2 2  7 ) 2 2  3 3 ) 3 2  9 ) 3 2  1 )  (x  (x  (x  (x  (x  (x  (x  (x  (x  x 6  10   2 x 2 x 3.  4 x  3  10 10 Exercises
4. 5. 6. 2 x  8 x  2 2 x  3 x  3  (x  (x  (x  (x 4 2  ) 4 2  ) 2 ) 2 3 2 3 ) 2  2 16 18  3 9 4 3 4  8 x  1  2 2 x Hint: Start by taking 2 out as though it were a common factor
4 2  ) 4 2  ) 2 ) 2 3 2 3 ) 2 4. 5. 6. 2 x  8 x  2 2 x  3 x  3 2 2 x  8 x  1  (x  (x 2  (x  (x  x 2  (x 2  (x 2  (x 2     1 2   4 2  7 7    x 4 2 2 ) 22 ) 2  2 )  2 16 18  9 4 3 4  3 2 1