1. Quadratic Functions(3) What is a perfect square. What is a perfect square. How to make and complete the square. How to make and complete the square. Sketching using completed square Sketching using completed square

2. A perfect square What do we get if we factorise: x2 + 10x + 25 This is called a perfect square because it can be written as (x+5)2. X+5 Can you think of an expression for a perfect cube??

3. Solving Quadratic Equations ► We will now look at solving quadratic equations using completing the square method.

4. Complete the square for: y = x 2 + 10x + 12 Use: (x + 5) 2 = x 2 + 10x + 25 x 2 + 10x + 12 = x 2 + 10x + 25 - 13 x 2 + 10x + 12 = (x + 5) 2 - 13 y = (x + 5) 2 - 13 … is complete square form 5 is half 10

5. Solve: x 2 + 10x + 12 = 0 (x + 5) 2 - 13 = 0 Solving Equations using the completed square Complete the square ….. (x + 5) 2 = 13 (x + 5) =  13 x = -5  13 x = -5 +  13 or -5 -  13 x = -1.39 or -8.61 The solutions SURD FORM (leave as square root)

6. Complete the square for: y = x 2 - 20x - 30 Use: (x - 10) 2 = x 2 - 20x + 100 x 2 - 20x - 30 = x 2 - 20x + 100 - 130 x 2 - 20x - 30 = (x - 10) 2 - 130 y = (x - 10) 2 - 130 … is completed square form -10 is half -20

7. Complete the square for: y = 2x 2 - 14x - 33 Use: (x - 3.5) 2 = x 2 - 7x + 12.25 x 2 - 7x - 16.5 = x 2 - 7x + 12.25 - 28.75 2(x 2 - 7x - 16.5) = 2( (x - 3.5) 2 - 28.75) y = 2( (x - 3.5) 2 - 28.75) … is complete square form -3.5 is half -7 Adjust to make a single ‘x 2’ : y = 2(x 2 - 7x - 16.5) y = 2 (x - 3.5) 2 – 57.5

8. Solve: 2x 2 - 14x - 33 = 0 Solving Equations using the completed square Complete the square (from previous slide)….. (x - 3.5) 2 = 28.75 (x - 3.5 ) =  28.75 x = 3.5  28.75 x = 3.5 +  28.75 or 3.5 -  28.75 x = 8.86 or -1.86 The solutions (x - 3.5) 2 - 28.75 = 0 x 2 - 7x – 16.5 = 0 (divide both sides by 2)

9. Quadratic graphs Investigate what happens when you change “a” and “b”.

10. Quadratic Graphs Investigate what happens when you change the value of k.

11. Quadratic graphs This is a translation of the graph y=kx 2 by the vector:

12. Finding critical values on graphs 1.Find the y-intercept 2.Find the x-intercept(s) 3.Find the vertex

13. Finding the y-intercept Intercepts y-axis when x=0

14. Finding the x-intercept(s) Intercepts x-axis when y=0 Does it factorise?? x=-2 and x=-8

15. Finding the vertex Find translation from y=x 2 by writing in completed square form. Vertex must be at (-5,-9)

16. Finding critical values on graphs 1.Find the y-intercept(0,16) 2.Find the x-intercept(s)(-2,0) & (-8,0) 3.Find the vertex(-5,-9) Now sketch this graph

17. Sketching the graph