Quadratics (Chapter 3) 95

11 A truck carrying a wide load needs to pass through the parabolic

tunnel shown. The units are metres.

The truck is 5 m high and 4 m wide.

a Find the quadratic function which describes the shape of the

tunnel.

b Determine whether the truck will fit.

12 Answer theOpening Problemon page 64.

The process of finding the maximum or minimum value of a function is calledoptimisation.

For the quadratic function y=ax^2 +bx+c, we have already seen that the vertex has

x-coordinate ¡

b

2 a

² If a> 0 , theminimumvalue ofyoccurs at x=¡

b

2 a

² If a< 0 , themaximumvalue ofyoccurs at x=¡

b

2 a

Example 28 Self Tutor

Find the maximum or minimum value of the following quadratic functions, and the corresponding

value ofx:

a y=x^2 +x¡ 3 b y=3+3x¡ 2 x^2

a y=x^2 +x¡ 3 has

a=1, b=1, and c=¡ 3.

Since a> 0 , the shape is

The minimum value occurs

when x=

¡b

2 a

=¡^12

and y=(¡^12 )^2 +(¡^12 )¡3=¡ (^314)
So, the minimum value ofyis¡ 314 ,
occurring when x=¡^12.
b y=¡ 2 x^2 +3x+3 has
a=¡ 2 , b=3, and c=3.
Since a< 0 , the shape is
The maximum value occurs
when x=
¡b
2 a

¡ 3
¡ 4
=^34
and y=¡2(^34 )^2 +3(^34 )+3=4^18
So, the maximum value ofyis 418 ,
occurring when x=^34.

H Quadratic optimisation

-3 3 x

y

8

O

ymin

x = -_^^b_____

2a_

ymax

b_____

x = -_^^2a_

4037 Cambridge

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