Quadratic Functions(3)Quadratic Functions(3)

•What is a perfect square.What is a perfect square.•How to make and complete the How to make and complete the

square.square.•Sketching using completed squareSketching using completed square

A perfect squareA perfect square

What do we get if we factorise:What do we get if we factorise:

xx22 + 10x + 25 + 10x + 25

This is called a perfect square because it can be written This is called a perfect square because it can be written as (x+5)as (x+5)22..

X+5

X+5

Can you think of an expression for a perfect

cube??

Solving Quadratic EquationsSolving Quadratic Equations

►We will now look at solving quadratic We will now look at solving quadratic equations using equations using completing the square completing the square method.method.

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 is half 10

Solve: x2 + 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 -8.61The solutions

SURD FORM(leave as square root)

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

-10 is half -20

Complete the square for: y = 2x2 - 14x - 33

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)

… is complete square form

-3.5 is half -7

Adjust to make a single ‘x2’ : y = 2(x2 - 7x - 16.5)

y = 2(x - 3.5)2 – 57.5

Solve: 2x2 - 14x - 33 = 0Solving 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

x2 - 7x – 16.5 = 0 (divide both sides by 2)

Quadratic graphsQuadratic graphs

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

baxy 2

Quadratic GraphsQuadratic Graphs

Investigate what happens when you change the value of k.

2kxy

Quadratic graphsQuadratic graphs

baxky 2

b

aThis is a translation of the graph y=kx2 by the vector:

Finding critical values on Finding critical values on graphsgraphs

16102 xxy

1.Find the y-intercept

2.Find the x-intercept(s)

3.Find the vertex

Finding the y-interceptFinding the y-intercept

16102 xxy

Intercepts y-axis when x=0

1601002 y

16y

Finding the x-intercept(s)Finding the x-intercept(s)

16102 xxy

Intercepts x-axis when y=0

016102 xx 082 xx

Does it factorise??

x=-2 and x=-8

Finding the vertexFinding the vertex

16102 xxy

Find translation from y=x2 by writing in completed square form.

95 2 xy

Vertex must be at (-5,-9)

Finding critical values on Finding critical values on graphsgraphs

16102 xxy

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

Sketching the graphSketching the graph