Cambridge Additional Mathematics

82 Quadratics (Chapter 3)

b The vertex is (^23 ,¡^13 )

and they-intercept is 1.

EXERCISE 3D.2

1 Write the following quadratics in the form y=(x¡h)^2 +k by ‘completing the square’. Hence

sketch each function, stating the coordinates of the vertex.

a y=x^2 ¡ 2 x+3 b y=x^2 +4x¡ 2 c y=x^2 ¡ 4 x

d y=x^2 +3x e y=x^2 +5x¡ 2 f y=x^2 ¡ 3 x+2

g y=x^2 ¡ 6 x+5 h y=x^2 +8x¡ 2 i y=x^2 ¡ 5 x+1

2 For each of the following quadratics:

i Write the quadratic in the formy=a(x¡h)^2 +k.

ii State the coordinates of the vertex.

iii Find they-intercept.

iv Sketch the graph of the quadratic.

a y=2x^2 +4x+5 b y=2x^2 ¡ 8 x+3

c y=2x^2 ¡ 6 x+1 d y=3x^2 ¡ 6 x+5

e y=¡x^2 +4x+2 f y=¡ 2 x^2 ¡ 5 x+3

QUADRATIC FUNCTIONS OF THE FORM y=ax^2 +bx+c

We now consider a method of graphing quadratics of the form y=ax^2 +bx+c directly, without having

to first convert them to a different form.

We know that the quadratic equation ax^2 +bx+c=0 has

solutions

¡b¡

p

¢

2 a

and

¡b+

p

¢

2 a

where ¢=b^2 ¡ 4 ac.

If ¢> 0 , these are thex-intercepts of the quadratic function

y=ax^2 +bx+c.

The average of the values is

¡b

2 a

, so we conclude that:

² the axis of symmetry is x=¡b

2 a

² the vertex of the quadratic hasx-coordinate

¡b

2 a

.

ais always the factor

to be ‘taken out’.

y

x

y = ax + bx + c 2

-b-~`

%%^%%

2a

¢ +b-~`

%%^%%

2a

¢

x=^^-b

2a

O

x

x=We

y

-0.5 1 1.5

1

-1

V,(-)We Qe

y= x - x+ 3412

O

cyan magenta yellow black

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100 4037 Cambridge
Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_03\082CamAdd_03.cdr Friday, 20 December 2013 12:37:00 PM GR8GREG