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Functions Quadratic Functions y = ax 2 Quadratics y = ax 2 +c Quadratic Functions Int 2 Quadratics y = a(x-b) 2 Quadratics y = a(x-b) 2 + c Factorised form y = (x-a)(x-b)

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Starter Int 2

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Learning Intention Success Criteria 1.Understand the term function. 1.To explain the term function. 2.Work out values for a given function. Int 2 Functions

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Int 2 Functions 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 =10A(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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Int 2 Functions Using the formula for the function we can make a table and draw a graph using A as the y coordinate.x012345A In the case The graph is a straight line We can this a Linear function.

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Int 2 Functions For the following functions write down the gradient and were the function crosses the y-axis f(x) = 2x - 1f(x) = 0.5x + 7f(x) = -3x Sketch the following functions. f(x) = xf(x) = 2x + 7f(x) = x +1

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Now try MIA Ex 1 Ch14 (page 216) Int 2 Functions

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Starter Int 2

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Learning Intention Success Criteria 1.To know the properties of a quadratic function. 1.To explain the main properties of the basic quadratic function y = ax 2 using graphical methods. 2.Understand the links between graphs of the form y = x 2 and y = ax 2 Int 2 Quadratic Functions

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Int 2 Quadratic Functions A function of the form f(x) = a x 2 + b x + c is called a quadratic function The simplest quadratics have the form f(x) = a x 2 Lets investigate

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Now try MIA Ex 2 Q2 P 219 Int 2 Quadratic Functions

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Quadratic of the form f(x) = ax 2 Key Features Symmetry about x =0 Vertex at (0,0) The bigger the value of a the steeper the curve. -x 2 flips the curve about x - axis

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Int 2 Quadratic Functions Example The parabola has the form y = ax 2 graph opposite. The point (3,36) lies on the graph. Find the equation of the function. Solution f(3) = = a x 9 a = 36 ÷ 9 a = 4 f(x) = 4x 2 (3,36)

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Now try MIA Ex 2 Q3 (page 219) Int 2 Quadratic Functions

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Int 2 Starter Q1. Write down the equation of the quadratic. Solution f(2) = = a x 4 a = 100 ÷ 4 a = 25 f(x) = 25x 2 (2,100) (x-4)(x-3) f(x) = ax 2

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Learning Intention Success Criteria 1.To know the properties of a quadratic function. y = ax 2 + c 1.To explain the main properties of the basic quadratic function y = ax 2 + c using graphical methods. 2.Understand the links between graphs of the form y = x 2 and y = ax 2 + c Int 2 Quadratic Functions

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Now try MIA Ex 2 Q5 (page 220) Int 2 Quadratic Functions Quadratic of the form f(x) = ax 2 + c

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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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Int 2 Quadratic Functions Example The parabola has the form y = ax 2 + 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 = a x 9 +2 a = (38 -2) ÷ 9 a = 4 f(x) = 4x (3,38) (0,2) f(x) = a x 2 + c

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Now try MIA Ex 2 Q7 (page 221) Int 2 Quadratic Functions

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Int 2 Starter Q1. Write down the equation of the quadratic. Solution f(9) = = a x 9 a = 81 ÷ 9 a = 9 f(x) = 9x 2 (9,81) (x-5)(x-6)

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Learning Intention Success Criteria 1.To know the properties 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.Understand the links between graphs of the form y = x 2 and y = a(x – b) 2 Int 2 Quadratic Functions

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Now try MIA Ex 3 Q2 (page 222) Int 2 Quadratic Functions Quadratic of the form f(x) = a(x - b) 2

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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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Int 2 Quadratic 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

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Now try MIA Ex 3 Q4 and Q5 (page 222) Int 2 Quadratic Functions

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Homework MIA Ex 4 (page 222) Int 2 Quadratic Functions

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Starter Int 2 (5,25) x f(x)

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Learning Intention Success Criteria 1.To know the properties 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.Understand the links between the graph of the form y = x 2 and y = a(x-b) 2 + c Int 2 Quadratic Functions

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Int 2 Quadratic 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 point a < 0 maximum turning point

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Int 2 Quadratic Functions Example 1 Sketch the graph y = (x - 3) (3,2) = (3,2) (0,11) Axis of symmetry at b = 3 = 11 a = 1 Vertex / turning point is (b,c) y = (0 - 3) b = 3 c = 2 x y y = a(x-b) 2 +c

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Int 2 Quadratic Functions Example2 Sketch the graph y = -(x + 2) (-2,1) = (-2,1) (0,-3) Axis of symmetry at b = -2 = -3 a = -1 Vertex / turning point is (b,c) y = -(0 + 2) b = -2 c = 1 x y y = a(x-b) 2 +c

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Int 2 Quadratic Functions Example Write down equation of the curve Given 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) y = a(x-b) 2 +c

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Now try MIA Ex 5 Q1 and Q2 (page 225) Int 2 Quadratic Functions

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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

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Now try MIA Ex6 (page 226) Int 2 Quadratic Functions

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Starter Int 2 (3,-6) x f(x)

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Learning Intention Success Criteria 1.To interpret the keyPoints of the factorised form of a quadratic function. 1.To show factorised form of a quadratic function. Int 2 Quadratic Functions

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Int 2 Quadratic Functions Some quadratic functions can be written in the factorised form y = (x - a)(x - b) The zeros / roots of this function occur when y = 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 form f(x) ax 2 + bx + c

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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 symmetry ALWAYS halfway between x = 2 and x = 4 x =3 Y – coordinate - turning point y = (3 - 2)(3 - 4) = -1 (3,-1)

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Now try MIA Ex7 (page 227) Int 2 Quadratic Functions

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