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