Page 10 - (1-1)eng

Elly fat mn 3omry ( 1 – 1 )
The Quadratic Function

2
 The general form of the quadratic function is ƒ ( x ) = a x + b x + c, a  0

2
This form can be written in the form ƒ ( x ) = a ( x – h ) + k

 b
 The corresponding graph is a parabola whose vertex is ( h, k ) where h = , k = ƒ ( h ),
2a

The parabola opens upwards if a > 0 and downwards if a < 0.

 The line x = h is the axis of symmetry of the parabola.

y y y y y y
x
x
Quadratic function (x) = a x 2 + b x + c (x) = x – 2 x – 3 (x) = – x + 2 x + 3 (x) = x – 2 x + 1 (x) = – x + 2 x – 1 (x) = x – 4 x + 5 (x) = – x + 4 x –5
x
x
x
x
2
2
2
2

2
2
2
2
2
b – 4 a c > 0 b – 4 a c = 0 b – 4 a c < 0

There are two different roots There are two equal roots There are no real roots

– 1 3 1
as a 0 Opposite a 0 as a as a 0 as a as a

The sign of the quadratic function x – 1 3 x – 1 3 x 1 x 1 x x
(x) + 0 – 0 + (x) – 0 + 0 – (x) + 0 + (x) – 0 – (x) + (x) –

We can choose a representative test value in each interval to check our work.

The Formula to Solve a Quadratic Equation in one Variable
2
2
a x + b x + c = 0 , a  0  x =  b  b  4 ac
2 a
Remarks

 b c
 The sum of the two roots = , The product of the two roots =
a a

2
 The quadratic equation whose roots are  ,  is x – ( +  ) x +   = 0
Calculus 9 Unit (1)