Functions Quadratic Functions y = ax 2 Quadratics y = ax 2 +c Quadratic Functions Int 2 Quadratics...
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Transcript of Functions Quadratic Functions y = ax 2 Quadratics y = ax 2 +c Quadratic Functions Int 2 Quadratics...
Functions
Quadratic Functions y = ax2
Quadratics y = ax2 +c
Quadratic FunctionsQuadratic Functionsw
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Int 2
Quadratics y = a(x-b)2
Quadratics y = a(x-b)2 + c
Factorised form y = (x-a)(x-b)
StarterStarter
2 4x +6x
Q3. Solve 3x + 1 = 19
1. Factorise the following.
2. Round to 3 sig. figures.
(a) 47856 (b) 0.065797 (c) 2.05700
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Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Understand the term Understand the term function.function.
1. To explain the term function.
2.2. Work out values for a Work out values for a given function.given function.
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Int 2
FunctionsFunctions
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Int 2FunctionsFunctions
A roll of carpet is 5m wide. It is solid in strips by the area.If the length of a strip is x m then the area. A square metres,is given by A = 5x.
A(x) =5x
Example
A(1) = 5 x 1 =5A(2) = 5 x 2 =10 A(t) = 5 x t = 5t
We say A is a function of x. We write :
The value of A depends on the value of x.
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Using the formula for the function we can make a table anddraw a graph using A as the y coordinate.
xx 00 11 22 33 44 55
AA 00 55 1010 1515 2020 2525
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y = A
(x) In the case
The graph is a straight line
We can this aLinear function.
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Int 2FunctionsFunctions
For the following functions write down the gradient and were the function crosses the y-axis
f(x) = 2x - 1 f(x) = 0.5x + 7 f(x) = -3x
Sketch the following functions.
f(x) = x f(x) = 2x + 7 f(x) = x +1
Now try MIA Ex 1Ch14 (page 216)
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Int 2
Caculate f (-1) , f (0) and f (2)
Q3. Solve 6x +1 = 55
21. Given the function f(x) = x
2. Round to 2 decimal places
(a) 47.856 (b) 0.065797 (c) 2.05500
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. To know the properties To know the properties of a quadratic function.of a quadratic function.
1. To explain the main properties of the basic quadratic function y = ax2
using graphical methods.
2.2. Understand the links Understand the links between graphs of the between graphs of the form y = xform y = x22 and and y = axy = ax22
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A function of the form
f(x) = a x2 + b x + c
is called a quadratic function
a, b and c
are constants
a 0
The simplest quadratics have the form
f(x) = a x2
Lets investigate
Now try MIA Ex 2Q2 P 219
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Quadratic FunctionsQuadratic Functions
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x2
3x2
5x2
x2
2 x2
x
Quadratic of the form f(x) = ax2
Key Features
Symmetry about x =0
Vertex at (0,0)
The bigger the value
of a the steeper the curve.
-x2 flips the curve about x - axis
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Int 2
Quadratic FunctionsQuadratic Functions
Example
The parabola has the form y = ax2 graph opposite. The point (3,36) lies on the graph.Find the equation of the function.
Solutionf(3) = 36
36 = a x 9
a = 36 ÷ 9
a = 4f(x) = 4x2
(3,36)
Now try MIA Ex 2Q3 (page 219)
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StarterStarter
Q1. Write down the equation of the quadratic.
Solutionf(2) = 100
100 = a x 4
a = 100 ÷ 4
a = 25f(x) = 25x2
(2,100)
2Q2. Factorise x - 7x +12 (x-4)(x-3)
f(x) = ax2
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. To know the properties To know the properties of a quadratic function.of a quadratic function.
y = ax2+ c
1. To explain the main properties of the basic quadratic function
y = ax2+ c using graphical
methods.2.2. Understand the links Understand the links
between graphs of the between graphs of the form y = xform y = x22 and and y = axy = ax2 2
+ c+ c
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Now try MIA Ex 2Q5 (page 220)
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Quadratic FunctionsQuadratic Functions
Quadratic of the form f(x) = ax2 + c
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x24
x21
3x21
x2 3
2 x22
x
Quadratic of the form f(x) = ax2 + c
Key Features
Symmetry about x = 0
Vertex at (0,C)
a > 0 the vertex (0,C) is a minimum turning point.
a < 0 the vertex (0,C) is a maximum turning point.
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Quadratic FunctionsQuadratic Functions
Example
The parabola has the form y = ax2 + c graph opposite. The vertex is the point (0,2) so c = 2. The point (3,38)lies on the graph. Find the equation of the function.
Solution
f(3) = a x 32 + 2
38 = a x 9 +2
a = (38 -2) ÷ 9
a = 4 f(x) = 4x2 + 2
(3,38)
(0,2)f(x) = a x2 + c
Now try MIA Ex 2Q7 (page 221)
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StarterStarter
Q1. Write down the equation of the quadratic.
Solutionf(9) = 81
81 = a x 9
a = 81 ÷ 9
a = 9f(x) = 9x2
(9,81)
2Q2. Factorise x - 11x +30 (x-5)(x-6)
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. To know the properties To know the properties of a quadratic function.of a quadratic function.
y = a(x – b)2
1. To explain the main properties of the basic quadratic function
y = a(x - b)2
using graphical methods.
2.2. Understand the links Understand the links between graphs of the between graphs of the form form
y = xy = x22 and and y = a(x – b)y = a(x – b)22
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Now try MIA Ex 3Q2 (page 222)
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Quadratic FunctionsQuadratic Functions
Quadratic of the form f(x) = a(x - b)2
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x 3( )2
x 4( )2
x 1( )2
2 x 3( )2
x 1( )2
x
Quadratic of the form f(x) = a(x - b)2
Key Features
Symmetry about x = b
Vertex at (b,0)Cuts y - axis at x =
0
a > 0 the vertex (b,0) is a minimum turning point.
a < 0 the vertex (b,0) is a maximum turning point.
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Quadratic FunctionsQuadratic Functions
Example
The parabola has the form f(x) = a(x – b)2. The vertex is the point (2,0) so b = 2. The point (5,36)lies on the graph. Find the equation of the function.
Solution
f(5) = a ( 5 - 2)2
36 = a x 9
a = 36 ÷ 9
a = 4 f(x) = 4(x-2)2
(5,36)
(2,0)
f(x) = a (x - b)2
Now try MIA Ex 3Q4 and Q5 (page
222)
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Homework MIA Ex 4 (page 222)
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2
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3 4 1 21. Calculate , +
c 2n x x
Q3. Given the f unction has the
f orm y = ax . Write down equation.
2. Make w the subject of the formula
y = k + 2w(5,25)
x
f(x)
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. To know the properties To know the properties of a quadratic function.of a quadratic function.
1. To explain the main properties of the basic quadratic function
y = a(x-b)2 + c using graphical
methods.
2.2. Understand the links Understand the links between the graph of between the graph of the form the form
y = xy = x22
and and
y = a(x-b)y = a(x-b)22 + c + c
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Quadratic FunctionsQuadratic Functions
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Quadratic FunctionsQuadratic Functions
Every quadratic function can be written in the form
y = a(x - b)2+c
axis of symmetry at x = b
Vertex or turning point at (b,c)
(b,c)
The curve y= f(x) is a parabola
x = b
Y - intercept
Cuts y-axis when x = 0 y = a(x – b)2 + c
a > 0 minimum turning pointa < 0 maximum turning point
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Int 2
Quadratic FunctionsQuadratic Functions
Example 1 Sketch the graph y = (x - 3)2 + 2
(3,2)
= (3,2)
(0,11)
Axis of symmetry at b = 3
= 11
a = 1
Vertex / turning point is (b,c)
y = (0 - 3)2 + 2
b = 3 c = 2
x
y
y = a(x-b)2+c
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Int 2
Quadratic FunctionsQuadratic Functions
Example2 Sketch the graph y = -(x + 2)2 + 1
(-2,1)
= (-2,1)
(0,-3)
Axis of symmetry at b = -2
= -3
a = -1
Vertex / turning point is (b,c)
y = -(0 + 2)2 + 1
b = -2 c = 1
x
y
y = a(x-b)2+c
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Example Write down equation of the curveGiven a = 1 or a = -1
(0,-4)
a = -1 (-3,5)
b = -3
c = 5
a < 0 maximum turning point
Vertex / turning point is (-3,5)
y = -(x + 3)2 + 5
y = a(x-b)2+c
Now try MIA Ex 5Q1 and Q2 (page
225)
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x 6( )23
x 5( )2 2
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Quadratic of the form f(x) = a(x - b)2 + c
a > 0 the vertex is a minimum.
a < 0 the vertex is a maximum.
Symmetry about x =b
Vertex / turning point at (b,c)
Cuts y - axis when x=0
Now try MIA Ex6 (page 226)
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2 2
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1 3 1 21. Calculate , +
2x y xy x
Q3. Given the f unction has the
f orm y = a(x -b) +c.
and is either 1 or -1.
a
2. Make k the subject of the formula
a = bk + 2d
Write down equation. (3,-6)
x
f(x)
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. To interpret the To interpret the keyPoints of the keyPoints of the factorised form of a factorised form of a quadratic function.quadratic function.
1. To show factorised form of a quadratic function.
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Some quadratic functions can be written in the factorised form
y = (x - a)(x - b)
The zeros / roots of this function occur wheny = 0 (x - a)(x - b) = 0 x = a and x = b
Note: The a,b in this form are NOT the a,b in the formf(x) ax2 + bx + c
Q. Find the zeros, axis of symmetry and turning point for f(x) = (x - 2)(x - 4)
Zero’s at x = 2 and x = 4
Axis of symmetryALWAYS halfway
between x = 2 and x = 4
x =3
Y – coordinate - turning point y = (3 - 2)(3 - 4) = -1(3,-1)
Now try MIA Ex7 (page 227)
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