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

ww

.math

srevis

ion

.com

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

ww

w.m

ath

srevis

ion

.com

Int 2

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.

ww

w.m

ath

srevis

ion

.com

Int 2

FunctionsFunctions

ww

w.m

ath

srevis

ion

.com

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.

ww

w.m

ath

srevis

ion

.com


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

0

5

10

15

20

25

30

0 2 4 6

x

y = A

(x) In the case

The graph is a straight line

We can this aLinear function.

ww

w.m

ath

srevis

ion

.com


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)

ww

w.m

ath

srevis

ion

.com


StarterStarterw

ww

.math

srevis

ion

.com

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


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

ww

w.m

ath

srevis

ion

.com

Int 2

Quadratic FunctionsQuadratic Functions

ww

w.m

ath

srevis

ion

.com

Int 2


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

ww

w.m

ath

srevis

ion

.com

Int 2


5 4 3 2 1 0 1 2 3 4 5

20

18

16

14

12

10

8

6

4

2

2

4

6

8

10

12

14

16

18

20

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

ww

w.m

ath

srevis

ion

.com

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.

Solutionf(3) = 36

36 = a x 9

a = 36 ÷ 9

a = 4f(x) = 4x2

(3,36)

Now try MIA Ex 2Q3 (page 219)

ww

w.m

ath

srevis

ion

.com

Int 2


ww

w.m

ath

srevis

ion

.com

Int 2

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



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

ww

w.m

ath

srevis

ion

.com

Int 2



ww

w.m

ath

srevis

ion

.com

Int 2


Quadratic of the form f(x) = ax2 + c

5 4 3 2 1 0 1 2 3 4 5

20

18

16

14

12

10

8

6

4

2

2

4

6

8

10

12

14

16

18

20

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.

ww

w.m

ath

srevis

ion

.com

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.

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


ww

w.m

ath

srevis

ion

.com

Int 2


ww

w.m

ath

srevis

ion

.com

Int 2

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)



y = a(x – b)2


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

ww

w.m

ath

srevis

ion

.com

Int 2



ww

w.m

ath

srevis

ion

.com

Int 2


Quadratic of the form f(x) = a(x - b)2

8 7 6 5 4 3 2 1 0 1 2 3 4 5 6

20

18

16

14

12

10

8

6

4

2

2

4

6

8

10

12

14

16

18

20

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.

ww

w.m

ath

srevis

ion

.com

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(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)

ww

w.m

ath

srevis

ion

.com

Int 2


Homework MIA Ex 4 (page 222)

ww

w.m

ath

srevis

ion

.com

Int 2


StarterStarterw

ww

.math

srevis

ion

.com

Int 2

2

2

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)




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

ww

w.m

ath

srevis

ion

.com

Int 2


ww

w.m

ath

srevis

ion

.com

Int 2


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

ww

w.m

ath

srevis

ion

.com

Int 2


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

ww

w.m

ath

srevis

ion

.com

Int 2


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

ww

w.m

ath

srevis

ion

.com

Int 2


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)

ww

w.m

ath

srevis

ion

.com

Int 2


10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8

20

18

16

14

12

10

8

6

4

2

2

4

6

8

10

12

14

16

18

20

x 3( )22

x 6( )23

x 5( )2 2

x

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)

ww

w.m

ath

srevis

ion

.com

Int 2


StarterStarterw

ww

.math

srevis

ion

.com

Int 2

2 2

2

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)


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.

ww

w.m

ath

srevis

ion

.com

Int 2


ww

w.m

ath

srevis

ion

.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 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)

ww

w.m

ath

srevis

ion

.com

Int 2


Date post:	31-Mar-2015
Category:	Documents
Upload:	darren-adkins
View:	231 times
Download:	0 times

Functions Quadratic Functions y = ax 2 Quadratics y = ax 2 +c Quadratic Functions Int 2 Quadratics...

Documents