Page 52 - Mathematics_Form_3

Functions

2 2
b _
b _
y y = a x + + c –
Maximum point ( 2a ) 4a
2 2
4ac − b
b _
= a x + + _

)
(
2a
4a
2
4ac − b
b _

y − _ 2 = a x +
(
)
(
)
4a
2a
FOR ONLINE READING ONLY
2
b _
Thus if a > 0, then a x + ≥ 0,
)
(
2a
4ac − b
x hence y ≥ _ 2 This means that, the

.
4a
function has a minimum value at
Axis of symmetry 4ac − b
2

y = _ , which is attained when
Figure 2.6: Graph of a quadratic function 4a
b _
b _
opening downwards x + = 0, that is, at x = − , in this
2a 2a
Graphs of quadratic functions are case, the graph opens upwards.
symmetric about the vertical line through
the minimum or maximum point called 2
b _
(
)
the axis of symmetry. Similarly, if a < 0, then a x + ≤ 0,
2a
2
4ac − b
.

A sketch of the graph of a function is a hence y ≤ _ This means that, the
4a
rough shape of a graph which includes function has a maximum value at
only the key features while a plot of y = _ 2 , which is attained when
4ac − b

the graph of a function is the accurate 4a
b _
b _
shape of a graph. x + = 0, that is, at x = − , in this
2a
2a
case, the graph opens downwards.
Properties of graphs of quadratic
functions b _
Consider the function The line given by x = − is the axis of
2a
f (x ) = a x + bx + c, where a, b and c are symmetry of the graph of a quadratic
2
constants and a ≠ 0. By completing the function. The point at which the
square, the function can be written as: quadratic function attains its maximum
or minimum value is called the turning
2
y = a x + bx + c,
point of the function and is given by; Mathematics for Secondary Schools
y
c _
b _
_
= x + x +  b 4ac b  − 2
2
a
a
a
y  − 2a , 4a  .
_
b _
c _
− = x + x  
2
a
a
a
y 2 2 b _
b _
_ c _ ( 2a ) 2 b _ ( 2a ) where x = − is the axis of symmetry
b _
− + = x + x +
2a
a
a
a
y 2 2 _ 2
4ac − b
b
c _
b _
_
_
− + = x + and y = 4a is either the maximum
)
(
a
a
4 a
2a
2
2 2 or the minimum value of the quadratic
b _
b _
y – c + = a x +
4a ( 2a ) function.
Student\s Book Form Three 45
18/09/2025 09:58:56
MATHEMATIC F3 SB.indd 45 18/09/2025 09:58:56
MATHEMATIC F3 SB.indd 45