Cambridge International Mathematics

EXERCISE 21F.2

1 Sketch the graph of the quadratic function with:

a x-intercepts¡ 1 and 1 , andy-intercept¡ 1 b x-intercepts¡ 3 and 1 , andy-intercept 2

c x-intercepts 2 and 5 , andy-intercept¡ 4 d x-intercept 2 andy-intercept 4.

2 Sketch the graphs of the following by considering:

i the value ofa ii they-intercept iii thex-intercepts.

a y=x^2 ¡ 4 x+4 b y=(x¡1)(x+3) c y=2(x+2)^2

d y=¡(x¡2)(x+1) e y=¡3(x+1)^2 f y=¡3(x¡4)(x¡1)

g y=2(x+ 3)(x+1) h y=2x^2 +3x+2 i y=¡ 2 x^2 ¡ 3 x+5

We have seen from the previous exercise that the graph of any quadratic function:

² is aparabola ² is symmetrical about aline of symmetry

² has aturning pointorvertex.

Discovery 3 Line of symmetry and vertex

#endboxedheading

What to do:

1 Use the quadratic formula to find the coordinates of A

and B.

2

3 The vertex of the parabola lies on the line of symmetry. By considering graphs for different values

ofa, discuss the values ofafor which the vertex of a quadratic is a maximum value or a minimum

value.

You should have discovered that:

the equation of theline of symmetryof y=ax^2 +bx+c is x=

¡b

2 a

.

This equation is true for all quadratic functions, not just those with twox-intercepts.

G LINE OF SYMMETRY AND VERTEX [3.2]

x

y xh=

@\=\$!X\+\%!\+\^

AB

dd

O

Consider the quadratic function y=ax^2 +bx+c whose

graph cuts thex-axis at A and B. Let the equation of the line

of symmetry be x=h.

Since A and B are the same distancedfrom the line of

symmetry,hmust be the average of thex-coordinates of

A and B. Use this property to find the line of symmetry

in terms ofa,bandc.

Quadratic equations and functions (Chapter 21) 441

If the graph has two -intercepts then the line of symmetry must be mid-way between them. We will use

this property in the following Discovery to establish an equation for the line of symmetry.

x

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Y:\HAESE\IGCSE01\IG01_21\441IGCSE01_21.CDR Thursday, 30 October 2008 10:49:14 AM PETER