Quadratic Functions Functions Quadratic Functions y = ax2 Int 2 Functions Quadratic Functions y = ax2 Quadratics y = ax2 +c www.mathsrevision.com Quadratics y = a(x-b)2 Quadratics y = a(x-b)2 + c Factorised form y = (x-a)(x-b)

Starter Int 2 www.mathsrevision.com

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

Functions www.mathsrevision.com Int 2 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. The value of A depends on the value of x. We say A is a function of x. We write : www.mathsrevision.com A(x) =5x Example A(1) = 5 x 1 =5 A(2) = 5 x 2 =10 A(t) = 5 x t = 5t

Functions x 1 2 3 4 5 A 10 15 20 25 www.mathsrevision.com Int 2 Using the formula for the function we can make a table and draw a graph using A as the y coordinate. x 1 2 3 4 5 A 10 15 20 25 www.mathsrevision.com In the case The graph is a straight line We can this a Linear function.

Functions www.mathsrevision.com Int 2 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. www.mathsrevision.com f(x) = x f(x) = 2x + 7 f(x) = x +1

Functions Int 2 Now try MIA Ex 1 Ch14 (page 216) www.mathsrevision.com

Starter Int 2 www.mathsrevision.com

Quadratic Functions www.mathsrevision.com Int 2 Learning Intention Success Criteria To explain the main properties of the basic quadratic function y = ax2 using graphical methods. To know the properties of a quadratic function. Understand the links between graphs of the form y = x2 and y = ax2 www.mathsrevision.com

Quadratic Functions www.mathsrevision.com A function of the form Int 2 A function of the form f(x) = a x2 + b x + c is called a quadratic function www.mathsrevision.com The simplest quadratics have the form f(x) = a x2 Lets investigate

Quadratic Functions Now try MIA Ex 2 Q2 P 219 www.mathsrevision.com Int 2 Now try MIA Ex 2 Q2 P 219 www.mathsrevision.com

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

Quadratic Functions www.mathsrevision.com Example Int 2 Example The parabola has the form y = ax2 graph opposite. The point (3,36) lies on the graph. Find the equation of the function. (3,36) Solution f(3) = 36 www.mathsrevision.com 36 = a x 9 a = 36 ÷ 9 a = 4 f(x) = 4x2

Quadratic Functions Now try MIA Ex 2 Q3 (page 219) Int 2 Now try MIA Ex 2 Q3 (page 219) www.mathsrevision.com

Starter www.mathsrevision.com Int 2 Q1. Write down the equation of the quadratic. f(x) = ax2 (2,100) Solution f(2) = 100 100 = a x 4 www.mathsrevision.com a = 100 ÷ 4 a = 25 f(x) = 25x2 (x-4)(x-3)

Quadratic Functions www.mathsrevision.com y = ax2+ c Int 2 Learning Intention Success Criteria To explain the main properties of the basic quadratic function y = ax2+ c using graphical methods. To know the properties of a quadratic function. y = ax2+ c www.mathsrevision.com Understand the links between graphs of the form y = x2 and y = ax2 + c

Quadratic Functions Now try MIA Ex 2 Q5 (page 220) Int 2 Now try MIA Ex 2 Q5 (page 220) www.mathsrevision.com Quadratic of the form f(x) = ax2 + c

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.

Quadratic Functions www.mathsrevision.com Example Int 2 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. (3,38) Solution f(x) = a x2 + c www.mathsrevision.com (0,2) f(3) = a x 32 + 2 38 = a x 9 +2 a = (38 -2) ÷ 9 a = 4 f(x) = 4x2 + 2

Quadratic Functions Now try MIA Ex 2 Q7 (page 221) Int 2 Now try MIA Ex 2 Q7 (page 221) www.mathsrevision.com

Starter www.mathsrevision.com Int 2 Q1. Write down the equation of the quadratic. (9,81) Solution f(9) = 81 81 = a x 9 www.mathsrevision.com a = 81 ÷ 9 a = 9 f(x) = 9x2 (x-5)(x-6)

Quadratic Functions www.mathsrevision.com y = a(x – b)2 Int 2 Learning Intention Success Criteria To explain the main properties of the basic quadratic function y = a(x - b)2 using graphical methods. To know the properties of a quadratic function. y = a(x – b)2 www.mathsrevision.com Understand the links between graphs of the form y = x2 and y = a(x – b)2

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

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.

Quadratic Functions www.mathsrevision.com Example Int 2 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(x) = a (x - b)2 (5,36) www.mathsrevision.com f(5) = a ( 5 - 2)2 (2,0) 36 = a x 9 a = 36 ÷ 9 a = 4 f(x) = 4(x-2)2

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

Quadratic Functions Homework MIA Ex 4 (page 222) www.mathsrevision.com Int 2 Homework MIA Ex 4 (page 222) www.mathsrevision.com

Starter Int 2 f(x) www.mathsrevision.com (5,25) x

Quadratic Functions www.mathsrevision.com Int 2 Learning Intention Success Criteria To explain the main properties of the basic quadratic function y = a(x-b)2 + c using graphical methods. To know the properties of a quadratic function. Understand the links between the graph of the form y = x2 and y = a(x-b)2 + c www.mathsrevision.com

Quadratic Functions y = a(x - b)2+c www.mathsrevision.com Int 2 Every quadratic function can be written in the form y = a(x - b)2+c The curve y= f(x) is a parabola axis of symmetry at x = b Y - intercept www.mathsrevision.com Vertex or turning point at (b,c) (b,c) Cuts y-axis when x = 0 y = a(x – b)2 + c x = b a > 0 minimum turning point a < 0 maximum turning point

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

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

Example Write down equation of the curve Quadratic Functions y = a(x-b)2+c Int 2 Example Write down equation of the curve Given a = 1 or a = -1 a < 0 maximum turning point a = -1 (-3,5) www.mathsrevision.com Vertex / turning point is (-3,5) b = -3 (0,-4) c = 5 y = -(x + 3)2 + 5

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

Quadratic of the form f(x) = a(x - b)2 + c Cuts y - axis when x=0 Symmetry about x =b Vertex / turning point at (b,c) a > 0 the vertex is a minimum. a < 0 the vertex is a maximum.

Quadratic Functions Now try MIA Ex6 (page 226) www.mathsrevision.com Int 2 Now try MIA Ex6 (page 226) www.mathsrevision.com

Starter Int 2 f(x) www.mathsrevision.com x (3,-6)

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

Quadratic Functions y = (x - a)(x - b) www.mathsrevision.com Int 2 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 www.mathsrevision.com Note: The a,b in this form are NOT the a,b in the form f(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 symmetry ALWAYS halfway between x = 2 and x = 4 x =3 Y – coordinate - turning point y = (3 - 2)(3 - 4) = -1 (3,-1)

Quadratic Functions Now try MIA Ex7 (page 227) www.mathsrevision.com Int 2 Now try MIA Ex7 (page 227) www.mathsrevision.com