# Www.mathsrevision.com Nat 5 Quadratic Functions Solving quadratic equations graphically Recap of Quadratic Functions / Graphs Factorising Methods for Trinomials.

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www.mathsrevision.com Nat 5 Quadratic Functions Solving quadratic equations graphically Recap of Quadratic Functions / Graphs Factorising Methods for Trinomials (Quadratics) Sketching a Parabola using Factorisation 28-Apr-15Created by Mr. Lafferty@mathsrevision.com Intersection points between a Straight Line and Quadratic Exam Type Questions Solving Quadratics by Factorising Solving Harder Quadratics by Factorising

28-Apr-15 Starter Questions www.mathsrevision.com Q1.Remove the brackets (x + 5)(x – 5) Q2.For the line y = -2x + 6, find the gradient and where it cuts the y axis. Nat 5 Created by Mr. Lafferty@mathsrevision.com Q3.A laptop costs £440 ( including @ 10% ) What is the cost before VAT.

28-Apr-15 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.Be able to create a coordinate grid. 1.We are learning how to sketch quadratic functions. 2.Be able to sketch quadratic functions. www.mathsrevision.com Quadratic Functions Nat 5

www.mathsrevision.com Nat 5 Quadratic Equations A quadratic function has the form f(x) = a x 2 + b x + c The graph of a quadratic function has the basic shape y x a, b and c are constants and a ≠ 0 y x a > 0 a < 0 The graph of a quadratic function is called a PARABOLA

28-Apr-15Created by Mr. Lafferty Maths Dept 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 y = x 2 xyxy -2 4 0 0 2 4 Quadratic Functions 3 9 -3 9 y = x 2 - 4 -2 0 0 -4 2 0 xyxy 3 5 -3 5 y = x 2 + x - 6 -2 -4 0 -6 2 0 xyxy 3 6 -4 6

28-Apr-15 Created by Mr. Lafferty@www.mathsrevision.com Now try N5 TJ Ex 14.1 Ch14 (page132) www.mathsrevision.com Factorising Methods Nat 5

Starter Questions 28-Apr-15Created by Mr. Lafferty@mathsrevision.com www.mathsrevision.com Q1.True or false y ( y + 6 ) -7y = y 2 -7y + 6 Q2.Fill in the ? 49 – 4x 2 = ( ? + ?x)(? – 2?) Q3.Write in scientific notation 0.0341 Nat 5

28-Apr-15 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.We are learning how to use the parabola graph to solve equations containing quadratic function. 1. Use graph to solve quadratic equations. www.mathsrevision.com Quadratic Functions Nat 5

www.mathsrevision.com Nat 5 Quadratic Equations A quadratic function has the form f(x) = a x 2 + b x + c The graph of a quadratic function has the basic shape The x-coordinates where the graph cuts the x – axis are called the Roots of the function. y x i.e. a x 2 + b x + c = 0 a, b and c are constants and a ≠ 0 y x This is called a quadratic equation

28-Apr-15Created by Mr. Lafferty Maths Dept Roots of a Quadratic Function x 2 – 11x + 28 = 0 x 2 + 5x = 0 0 123456 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 7 Find the solution of Find the solution of Graph of y = x 2 - 11x + 28 From the graph, setting y = 0 we can see that x = 4 and x = 7 Graph of y = x 2 + 5x From the graph, setting y = 0 we can see that x = -5 and x = 0

28-Apr-15 Created by Mr. Lafferty@www.mathsrevision.com Now try N5 TJ Ex 14.2 Ch14 (page133) www.mathsrevision.com Factorising Methods Nat 5

Starter Questions 28-Apr-15Created by Mr. Lafferty@mathsrevision.com www.mathsrevision.com In pairs and if necessary use notes to Write down the three types of factorising and give an example of each. Nat 5

28-Apr-15 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.To be able to identify the three methods of factorising. 1.We are reviewing the three basic methods for factorising. 2.Apply knowledge to problems. www.mathsrevision.com Factorising Methods Nat 5

28-Apr-15 Created by Mr. Lafferty@www.mathsrevision.com www.mathsrevision.com The main reason we learn the process of factorising is that it helps to solve (find roots) quadratic equations. 1.Take any common factors out and put them outside the brackets. 2.Check for the difference of two squares. 3.Factorise any quadratic expression left. Nat 5 Factors and Solving Quadratic Equations Reminder of Methods

28-Apr-15Created by Mr. Lafferty www.mathsrevision.com Type 1 : Taking out a common factor. (a) w 2 – 2w (b) 9b – b 2 (c)20ab 2 + 24a 2 b (d) 8c - 12c 2 + 16c 3 w( w – 2 ) b( 9 – b ) 4ab( 5b + 6a) Difference of Two Squares Nat 5 4c( 2 – 3c + 4c 2 )

28-Apr-15Created by Mr. Lafferty@www.mathsrevision.com www.mathsrevision.com When we have the special case that an expression is made up of the difference of two squares then it is simple to factorise The format for the difference of two squares a 2 – b 2 First square term Second square term Difference Difference of Two Squares Nat 5

28-Apr-15Created by Mr. Lafferty@www.mathsrevision.com www.mathsrevision.com a 2 – b 2 First square term Second square term Difference This factorises to ( a + b )( a – b ) Two brackets the same except for + and a - Check by multiplying out the bracket to get back to where you started Difference of Two Squares Nat 5

28-Apr-15Created by Mr. Lafferty www.mathsrevision.com Type 2 : Factorise using the difference of two squares (a) w 2 – z 2 (b) 9a 2 – b 2 (c)16y 2 – 100k 2 ( w + z )( w – z ) ( 3a + b )( 3a – b ) ( 4y + 10k )( 4y – 10k ) Difference of Two Squares Nat 5

28-Apr-15Created by Mr. Lafferty www.mathsrevision.com Factorise these trickier expressions. (a)6x 2 – 24 (b) 3w 2 – 3 (c) 8 – 2b 2 (d) 27w 2 – 12 6(x + 2 )( x – 2 ) 3( w + 1 )( w – 1 ) 2( 2 + b )( 2 – b ) 3(3 w + 2 )( 3w – 2 ) Difference of Two Squares Nat 5

www.mathsrevision.com Nat 5 + 1 + 2+ 2 28-Apr-15Created by Mr. Lafferty@mathsrevision.com x 2 + 3x + 2 Type 3 :Strategy for factorising quadratics Factorising Using St. Andrew’s Cross method x x+ 1 Find two numbers that multiply to give last number (+2) and Diagonals sum to give middle value +3x. ( ) x x (+2) x ( +1) = +2 (+2x) +( +1x) = +3x

www.mathsrevision.com Nat 5 + 5 + 1 + 5 28-Apr-15Created by Mr. Lafferty@mathsrevision.com x 2 + 6x + 5 Strategy for factorising quadratics Factorising Using St. Andrew’s Cross method x x+ 1 ( ) x x Find two numbers that multiply to give last number (+5) and Diagonals sum to give middle value +6x. (+5) x ( +1) = +5 (+5x) +( +1x) = +6x

www.mathsrevision.com Nat 5 - 2 - 2- 2 - 2 28-Apr-15Created by Mr. Lafferty@mathsrevision.com x 2 - 4x + 4 Strategy for factorising quadratics Factorising Using St. Andrew’s Cross method x x ( ) x x Both numbers must be - Find two numbers that multiply to give last number (+4) and Diagonals sum to give middle value -4x. (-2) x ( -2) = +4 (-2x) +( -2x) = -4x

www.mathsrevision.com Nat 5 - 3- 3 + 1 28-Apr-15Created by Mr. Lafferty@mathsrevision.com x 2 - 2x - 3 Strategy for factorising quadratics Factorising Using St. Andrew’s Cross method x x+ 1 ( ) x x One number must be + and one - Find two numbers that multiply to give last number (-3) and Diagonals sum to give middle value -2x (-3) x ( +1) = -3 (-3x) +( x) = -2x

www.mathsrevision.com Nat 5 - 4 3x - 4 + 1 28-Apr-15Created by Mr. Lafferty@mathsrevision.com 3x 2 - x - 4 Strategy for factorising quadratics Factorising Using St. Andrew’s Cross method 3x x+ 1 ( ) x One number must be + and one - Find two numbers that multiply to give last number (-4) and Diagonals sum to give middle value -x (-4) x ( +1) = -4 (3x) +( -4x) = -x

www.mathsrevision.com Nat 5 - 3 2x - 3 2x + 1 28-Apr-15Created by Mr. Lafferty@mathsrevision.com 2x 2 - x - 3 Strategy for factorising quadratics Factorising Using St. Andrew’s Cross method x+ 1 ( ) x One number must be + and one - Find two numbers that multiply to give last number (-3) and Diagonals sum to give middle value -x (-3) x ( +1) = -3 (-3x) +( +2x) = -x

www.mathsrevision.com Nat 5 4x 2 - 4x - 3 Two numbers that multiply to give last number (-3) and Diagonals sum to give middle value (-4x) 4x 28-Apr-15Created by Mr. Lafferty@mathsrevision.com Factorising Using St. Andrew’s Cross method x ( ) Keeping the LHS fixed Can we do it ! one number is + and one number is - Factors 1 and -3 -1 and 3

www.mathsrevision.com Nat 5 - 3 + 1 2x - 3 4x 2 - 4x - 3 2x 28-Apr-15Created by Mr. Lafferty@mathsrevision.com Factorising Using St. Andrew’s Cross method 2x ( ) Find another set of factors for LHS Factors 1 and -3 -1 and 3 Repeat the factors for RHS to see if it factorises now + 1 2x

28-Apr-15Created by Mr. Lafferty www.mathsrevision.com Factorise using SAC method (a)m 2 + 2m +1 (b) y 2 + 6m + 5 (c) 2b 2 + b - 1 (d)3a 2 – 14a + 8 (m + 1 )( m + 1 ) ( y + 5 )( y + 1 ) ( 2b - 1 )( b + 1 ) ( 3a - 2 )( a – 4 ) Factorising Using St. Andrew’s Cross method Nat 5

28-Apr-15 Created by Mr. Lafferty@www.mathsrevision.com Now try N5 TJ Ex 14.3 Ch14 (page134) www.mathsrevision.com Factorising Methods Nat 5

28-Apr-15 Starter Questions www.mathsrevision.com Q1.Multiple out the brackets and simplify. (a)( 2x – 5 )( x + 5 ) Created by Mr. Lafferty@mathsrevision.com Q3.True or false the gradient of the line is 1 x = y + 1 Q2.Find the volume of a cylinder with height 6m and diameter 9cm Nat 5

28-Apr-15 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.To be able to factorise. 1.We are learning how to solve quadratics by factorising. 2.Solve quadratics. www.mathsrevision.com Factorising Methods Nat 5

28-Apr-15 Created by Mr. Lafferty@www.mathsrevision.com www.mathsrevision.com Solving Quadratic Equations Nat 5 Examples Solve ( find the roots ) for the following x(x – 2) = 0 x = 0and x - 2 = 0 x = 2 4t(3t + 15) = 0 4t = 0and3t + 15 = 0 t = -5t = 0and

28-Apr-15 Created by Mr. Lafferty@www.mathsrevision.com www.mathsrevision.com Solving Quadratic Equations Nat 5 Examples Solve ( find the roots ) for the following x 2 – 4x = 0 x(x – 4) = 0 x = 0and x - 4 = 0 x = 4 16t – 6t 2 = 0 2t(8 – 3t) = 0 2t = 0and8 – 3t = 0 t = 8/3t = 0and Common Factor Common Factor

28-Apr-15 Created by Mr. Lafferty@www.mathsrevision.com www.mathsrevision.com Solving Quadratic Equations Nat 5 Examples Solve ( find the roots ) for the following x 2 – 9 = 0 (x – 3)(x + 3) = 0 x = 3and x = -3 100s 2 – 25 = 0 25(2s – 1)(2s + 1) = 0 2s – 1 = 0and2s + 1 = 0 s = - 0.5s = 0.5and Difference 2 squares Difference 2 squares Take out common factor 25(4s 2 - 1) = 0

www.mathsrevision.com Solving Quadratic Equations Nat 5 Examples 2x 2 – 8 = 0 2(x 2 – 4) = 0 x = 2andx = - 2 80 – 125e 2 = 0 5(16 – 25e 2 ) = 0 4 – 5e = 0and4 + 5t = 0 e = - 4/5e = 4/5and Common Factor Common Factor Difference 2 squares 2(x – 2)(x + 2) = 0 (x – 2)(x + 2) = 0 Difference 2 squares 5(4 – 5e)(4 + 5e) = 0 (4 – 5e)(4 + 5e) = 0 (x – 2) = 0and(x + 2) = 0

28-Apr-15 Created by Mr. Lafferty@www.mathsrevision.com Now try N5 TJ Ex 14.4 upto Q10 Ch14 (page135) www.mathsrevision.com Factorising Methods Nat 5

www.mathsrevision.com Solving Quadratic Equations Nat 5 Examples Solve ( find the roots ) for the following x 2 + 3x + 2 = 0 (x + 2)(x + 1) = 0 x = - 2andx = - 1 SAC Method x x 2 1 x + 2 = 0x + 1 = 0and 3x 2 – 11x - 4 = 0 (3x + 1)(x - 4) = 0 x = - 1/3andx = 4 SAC Method 3x x + 1 - 4 3x + 1 = 0andx - 4 = 0

www.mathsrevision.com Solving Quadratic Equations Nat 5 Examples Solve ( find the roots ) for the following x 2 + 5x + 4 = 0 (x + 4)(x + 1) = 0 x = - 4andx = - 1 SAC Method x x 4 1 x + 4 = 0x + 1 = 0and 1 + x - 6x 2 = 0 (1 + 3x)(1 – 2x) = 0 x = - 1/3andx = 0.5 SAC Method 1 1 +3x -2x 1 + 3x = 0and1 - 2x = 0

28-Apr-15 Created by Mr. Lafferty@www.mathsrevision.com Now try N5 TJ Ex 14.4 Q11.... Ch14 (page137) www.mathsrevision.com Factorising Methods Nat 5

www.mathsrevision.com Nat 5 28-Apr-15Created by Mr. Lafferty44 Starter Questions Q1.Round to 2 significant figures Q2.Why is 2 + 4 x 2 = 10 and not 12 Q3.Solve for x (a)52.567(b)626

28-Apr-15 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.Know the various methods of factorising a quadratic. 1.We are learning to sketch quadratic functions using factorisation methods. www.mathsrevision.com Nat 5 Sketching Quadratic Functions 2. Identify axis of symmetry from roots. 3. Be able to sketch quadratic graph.

www.mathsrevision.com Nat 5 Sketching Quadratic Functions We can use a 4 step process to sketch a quadratic function Example 2 : Sketch f(x) = x 2 - 7x + 6 Step 1 : Find where the function crosses the x – axis. i.e. x 2 – 7x + 6 = 0 SAC Method (x - 6)(x - 1) = 0 x = 6x = 1 x x - 6 - 1 x - 6 = 0x - 1 = 0 (6, 0)(1, 0)

www.mathsrevision.com Nat 5 Step 2 : Find equation of axis of symmetry. It is half way between points in step 1 Sketching Quadratic Functions Equation is x = 3.5 Step 3 : Find coordinates of Turning Point (TP) For x = 3.5 f(3.5) = (3.5) 2 – 7 x (3.5) + 6 = -6.25 Turning point TP is a Minimum at (3.5, -6.25) (6 + 1) ÷ 2 =3.5

www.mathsrevision.com Nat 5 (3.5,-6.25) Cuts x - axis at 1 and 6 6 1 Step 4 :Find where curve cuts y-axis. For x = 0f(0) = 0 2 – 7 x 0 = 6 = 6 (0,6) X Y Cuts y - axis at 6 Mini TP (3.5,-6.25) 6 Sketching Quadratic Functions Now we can sketch the curve y = x 2 – 7x + 6

www.mathsrevision.com Nat 5 Sketching Quadratic Functions We can use a 4 step process to sketch a quadratic function Example 1 : Sketch f(x) = 15 – 2x – x 2 Step 1 : Find where the function crosses the x – axis. i.e. 15 - 2x - x 2 = 0 SAC Method (5 + x)(3 - x) = 0 x = - 5x = 3 5 3 x - x 5 + x = 03 - x = 0 (- 5, 0)(3, 0)

www.mathsrevision.com Nat 5 Step 2 : Find equation of axis of symmetry. It is half way between points in step 1 Sketching Quadratic Functions Equation is x = -1 Step 3 : Find coordinates of Turning Point (TP) For x = -1f(-1) = 15 – 2 x (-1) – (-1) 2 = 16 Turning point TP is a Maximum at (-1, 16) (-5 + 3) ÷ 2 = -1

www.mathsrevision.com Nat 5 3 -5 Step 4 :Find where curve cuts y-axis. For x = 0f(0) = 15 – 2 x 0 – 0 2 = 15 (0,15) X Y Cuts x-axis at -5 and 3 Cuts y-axis at 15 Max TP (-1,16)(-1,16) 15 Sketching Quadratic Functions Now we can sketch the curve y = 15 – 2x – x 2

Quadratic Functions y = ax 2 + bx + c SAC e.g. (x+1)(x-2)=0 Graphs Evaluating Factorisation ax 2 + bx + c = 0 Roots x = -1 and x = 2 Roots Mini. Point (0, ) (0, ) Max. Point Line of Symmetry half way between roots Line of Symmetry half way between roots a > 0 a < 0 f(x) = x 2 + 4x + 3 f(-2) =(-2) 2 + 4 x (-2) + 3 = -1 x = x = cc

28-Apr-15 Created by Mr. Lafferty@www.mathsrevision.com Now try N5 TJ Ex 14.5 Ch14 (page138) www.mathsrevision.com Factorising Methods Nat 5

created by Mr. Lafferty Starter Questions www.mathsrevision.com Nat 5

created by Mr. Lafferty Learning Intention Success Criteria 1.Know how to rearrange and factorise a quadratic. 1.We are learning about intersection points between quadratics and straight lines. www.mathsrevision.com Nat 5 Intersection Points between Quadratics and Straight Line.

Intersection Points Between two lines Simultaneous Equations Make them equal to each other Rearrange into = 0 and then solve Between a line and a curve

Make them equal to each other Rearrange into … = 0 Find the intersection points between a line and a curve Example: y = x 2 y = x x 2 = x x 2 - x = 0 x ( x - 1) = 0 x = 0x = 1 Factorise Substitute x = 0 and x = 1 into straight line equation x = 0 y = 0 x = 1 y = 1 Intersection points ( 0, 0 ) and ( 1, 1 ) ( 0, 0 ) ( 1, 1 ) solve

Intersection points ( 1, 6 ) and ( 4, 3 ) ( 4, 3 ) ( 1, 6 ) Make them equal to each other Rearrange into … = 0 Find the intersection points between a line and a curve Example: y = x 2 – 6x + 11 y = -x + 7 x 2 - 6x + 11 = - x + 7 x 2 - 5x + 4 = 0 ( x - 1) (x – 4) = 0 x = 1x = 4 Factorise Substitute x = 1 and x = 4 into straight line equation x = 1 y = 6 x = 4 y = 3 solve

28-Apr-15 Created by Mr. Lafferty@www.mathsrevision.com Now try N5 TJ Ex 14.6 Ch14 (page139) www.mathsrevision.com Factorising Methods Nat 5