Cambridge International Mathematics

Example 24 Self Tutor

Find the quadratic function with:

a vertex(1,2)andy-intercept 3 b vertex(2,11)which passes through the point(¡ 1 ,¡7).

a As h=1and k=2,

f(x)=a(x¡1)^2 +2

But f(0) = 3

) a(¡1)^2 +2=3

) a+2=3

) a=1

So, f(x)=(x¡1)^2 +2

or f(x)=x^2 ¡ 2 x+3

fin expanded formg

b As h=2and k=11,

f(x)=a(x¡2)^2 +11

But f(¡1) =¡ 7

) a(¡3)^2 +11=¡ 7

) 9 a+11=¡ 7

) 9 a=¡ 18

) a=¡ 2

So, f(x)=¡2(x¡2)^2 +11

or f(x)=¡2(x^2 ¡ 4 x+4)+11

=¡ 2 x^2 +8x+3

Example 25 Self Tutor

The graph of a quadratic function hasx-intercepts¡^52 and^13 , and it passes

through(1,42). Find the function.

Thex-intercept¡^52 comes from the linear factor x+^52 or 2 x+5

Thex-intercept^13 comes from the linear factor x¡^13 or 3 x¡ 1

) f(x)=a(2x+ 5)(3x¡1)

But f(1) = 42,soa(7)(2) = 42

) 14 a=42

) a=3

Thus f(x) = 3(2x+ 5)(3x¡1)

EXERCISE 21H

In questions 1

and 2 we are

given that

a=1.

1 If f(x)=x^2 +bx+c, find f(x) given that its vertex is at:

a (1,3) b (0,2) c (3,0) d (¡ 3 ,¡2) e (^12 ,1)

2 If f(x)=x^2 +bx+c, find f(x) given that it hasx-intercepts:

a 0 and 2 b ¡ 4 and 1 c ¡ 5 and 2 d ¡ 7 and 0.

H FINDING A QUADRATIC FUNCTION [3.3, 3.4]

.

.

Having studied the graphs and properties of quadratics in some detail, we should be able to use facts about

a graph to determine the corresponding function.

Quadratic equations and functions (Chapter 21) 445

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Y:\HAESE\IGCSE01\IG01_21\445IGCSE01_21.CDR Monday, 27 October 2008 2:10:02 PM PETER