Cambridge Additional Mathematics

Example 15 Self Tutor

Write y=x^2 +4x+3 in the form y=(x¡h)^2 +k by ‘completing the square’.

Hence sketch y=x^2 +4x+3, stating the coordinates of the vertex.

y=x^2 +4x+3

) y=x^2 +4x+2^2 +3¡ 22

) y=(x+2)^2 ¡ 1

shift 2

units left

shift 1

unit down

The vertex is(¡ 2 ,¡1)

and they-intercept is 3.

Example 16 Self Tutor

a Converty=3x^2 ¡ 4 x+1to the formy=a(x¡h)^2 +k.

b Hence, write down the coordinates of its vertex and sketch the quadratic.

a y=3x^2 ¡ 4 x+1

=3[x^2 ¡^43 x+^13 ] ftaking out a factor of 3 g

=3[x^2 ¡2(^23 )x+(^23 )^2 ¡(^23 )^2 +^13 ] fcompleting the squareg

= 3[(x¡^23 )^2 ¡^49 +^39 ] fwriting as a perfect squareg

= 3[(x¡^23 )^2 ¡^19 ]

=3(x¡^23 )^2 ¡^13

-1

-1

-2

-2

y=x 2

y=x +4x+3 2

vertex(-2 -1),

y

O x

SKETCHING GRAPHS BY ‘COMPLETING THE SQUARE’

If we wish to graph a quadratic given in general form y=ax^2 +bx+c, one approach is to convert it to

the form y=a(x¡h)^2 +k where we can read off the coordinates of the vertex (h,k). To do this, we

‘complete the square’.

Consider the simple case y=x^2 ¡ 6 x+7,

for which a=1.

y=x^2 ¡ 6 x+7

) y=x^2 ¡ 6 x+3^2 +7¡ 32

) y=(x¡3)^2 ¡ 2

So, the vertex is (3,¡2).

To obtain the graph of y=x^2 ¡ 6 x+7 from the

graph of y=x^2 , we shift it 3 units to the right and

2 units down.

y

x

+3

-2

+3

O -2

y=x -6x+7 2

y=x 2

7

(3 -2),

Quadratics (Chapter 3) 81

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_03\081CamAdd_03.cdr Friday, 3 January 2014 11:48:09 AM GR8GREG