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Understanding Quadratic Functions and How to Complete the Square

Learn about perfect squares, solving quadratic equations, completing the square, and graphing quadratic functions step by step. Explore properties of perfect cubes and solve quadratic equations with SURD form. Practice finding critical values on graphs.

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Understanding Quadratic Functions and How to Complete the Square

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

  2. X+5 X+5 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. 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. 5 is half 10 Complete the square for: y = x2 + 10x + 12 Use: (x + 5)2 = x2 + 10x + 25 x2 + 10x + 12 = x2 + 10x + 25 - 13 x2 + 10x + 12 = (x + 5)2 - 13 y = (x + 5)2 - 13 … is complete square form

  5. Solving Equations using the completed square SURD FORM (leave as square root) Solve: x2 + 10x + 12 = 0 Complete the square ….. (x + 5)2 - 13 = 0 (x + 5)2 = 13 (x + 5)=13 x = -513 x = -5+ 13 or -5- 13 x =-1.39 or -8.61 The solutions

  6. -10 is half -20 Complete the square for: y = x2 - 20x - 30 Use: (x - 10)2 = x2 - 20x + 100 x2 - 20x - 30 = x2 - 20x + 100 - 130 x2 - 20x - 30 = (x - 10)2 - 130 y = (x - 10)2 - 130 … is completed square form

  7. Complete the square for: y = 2x2 - 14x - 33 -3.5 is half -7 Adjust to make a single ‘x2’ :y = 2(x2 - 7x - 16.5) Use: (x - 3.5)2 = x2 - 7x + 12.25 x2 - 7x - 16.5 = x2 - 7x + 12.25 - 28.75 2(x2 - 7x - 16.5) = 2((x - 3.5)2 - 28.75) y = 2((x - 3.5)2 - 28.75) y = 2(x - 3.5)2 – 57.5 … is complete square form

  8. Solving Equations using the completed square Solve: 2x2 - 14x - 33 = 0 x2 - 7x – 16.5 = 0 (divide both sides by 2) Complete the square (from previous slide)….. (x - 3.5)2 - 28.75= 0 (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

  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=kx2 by the vector:

  12. Finding critical values on graphs • Find the y-intercept • Find the x-intercept(s) • 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=x2 by writing in completed square form. Vertex must be at (-5,-9)

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

  17. Sketching the graph

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