Quadratic Functions

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Quadratic Functions
I. A quantity y is a quadratic function of x if it can be written in the form
y  f ( x)  ax 2  bx  c , where a, b and c are constants and a  0
 The expression ax 2  bx  c is a quadratic expression in standard form
 The term ax 2 is called the squared term or quadratic term
 The term bx is called the linear term.
 The term c is called the constant term.
II. A quadratic function in x is expressed in factored form if it is written as
y  f ( x)  a( x  r )( x  s ) , where a, r and s are constants and a  0
 The constants r and s are zeros of the function f ( x)  a( x  r )( x  s ) .
 The constant a is the leading coefficient, the same as the constant a in the standard form.
III. A quadratic function in x is expressed in vertex form if it is written as
y  f ( x)  a( x  h)2  k , where a, h and k are constants and a  0
For the function f ( x)  a( x  h)2  k
 f (h)  k , and the point ( h, k ) is the vertex of the graph.
 The constant a is the leading coefficient, the same as the constant a in the standard form
o If a  0 then k is the minimum value of the function and the graph opens upward.
o If a  0 then k is the maximum value of the function and the graph opens
downwards.
IV. To find the vertex of a quadratic function in standard form.
y  f ( x)  ax 2  bx  c
b
 To find the vertex x coordinate: x 
2a
 b
 b  
 Vertex  , f   
 2a  
 2a
V. Quadratic Formula
The quadratic formula combines both solutions of ax 2  bx  c  0
x
b  b2  4ac
2a
VI. Discriminant
For the equation ax 2  bx  c  0 , we define the discriminant D  b 2  4ac
 If D  b 2  4ac >0, the discriminant is positive, then the quadratic equation has two
distinct solutions (roots or zeros)
 If D  b 2  4ac = 0, the discriminate is zero, then the quadratic equation has only one
solution. This is sometimes referred to as a repeated root.
 If D  b 2  4ac <0, the discriminant is negative, then the quadratic equation has no real
solutions.
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