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Any function of the form
y = f (x) = ax 2 + bx + c
where a  0 is called a Quadratic Function
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Example:
y = 3x
a
= 3,
Note that if
function
2
- 2x + 1
b = -2, c = 1
a=0
we simply have the linear
y = bx + c
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Consider the simplest quadratic equation
y=x2
Here a = 1, b = 0, c = 0
Plotting some ordered pairs (x, y) we have:
y = f (x ) = x 2
x
-3
-2
-1
0
1
2
3
f (x )
9
4
1
0
1
4
9
(x, y )
(-3, 9)
(-2, 4)
(-1, 1)
(0, 0)
(1, 1)
(2, 4)
(3, 9)
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(x, y)
(-2, 4)
4
(-3, 9)
(-2, 4)
(-1, 1)
(0, 0)
(1, 1)
(2, 4)
y
(2, 4)
3
2
y = x2
1
(3, 9)
-3
-2
-1
Vertex (0, 0)
1
2
3 x
A parabola with the y-axis as the axis of symmetry.
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Graphs of y = ax
2 will have similar form and
the value of the coefficient ‘a ’ determines the
graph’s shape.
y
y = 2x 2
y=x
4
3
2
y = 1 /2 x 2
2
a>0
opening up
-3
-2
1
-1
1
2
3 x
6
y
x
a<0
opening down
y = -2x
2
In general the quadratic term ax 2 in the
quadratic function f (x ) = ax 2 +bx + c
determines the way the graph opens.
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Consider f (x ) = ax 2 +bx + c
In a general sense the linear term bx acts to shift
the plot of f (x ) from side to side and the constant
term c (=cx 0) acts to shift the plot up or down.
a>0
Notice that c is
the y -intercept
where x = 0 and
f (0) = c
y
x-intercept
c
a<0
c
x
y-intercept
Note also that the x -intercepts (if they exist)
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are obtained by solving: y = ax 2 +bx + c = 0
It turns out that the details of a quadratic
function can be found by considering its
coefficients a, b and c as follows:
(1) Opening up (a > 0), down (a < 0)
(2) y –intercept: c
(3) x -intercepts from solution of
y = ax
2
+ bx + c = 0
You solve by factoring or
the quadratic formula
- b - b 
(4) vertex =  ,f   
 2a  2a  
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Example:
y = f (x ) = x
2
-x-2
here a = 1, b = -1 and c = -2
(1) opens upwards since a > 0
(2) y –intercept: -2
(3) x -intercepts from x 2 - x - 2 = 0
or (x -2)(x +1) = 0
x = 2 or x = -1
-b
(4) vertex: h 
2a
 1  1 
 , f  
 2  2 
1
1
  , -2 
4
2
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y
(-1, 0)
-2
(2, 0)
-1
0
-1
-2
1
2
y=x
2
x
-x-2
1
1
 , -2 
4
2
-3
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Example:
y = j (x ) = x
2
-9
here a = 1, b = 0 and c = -9
(1) opens upwards since a > 0
(2) y –intercept: -9
(3) x -intercepts from x 2 - 9 = 0
or x 2 = 9  x = 3
(4) vertex at (0, -9)
- b
- b 

, j
 
 2a  
 2a
12
y
(-3, 0)
-3
(3, 0)
0
y=x
-9
x
3
2
-9
(0, -9)
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Example:
y = g (x ) = x 2 - 6x + 9
here a = 1, b = -6 and c = 9
(1) opens upwards since a > 0
(2) y –intercept: 9
(3) x -intercepts from x 2 - 6x + 9 = 0
or (x - 3)(x - 3) = 0  x = 3 only
(4) vertex:
-b
h
2a
3, g 3 
 3, 0 
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y
9
(0, 9)
y=x
(3, 0)
3
2
- 6x + 9
x
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Example:
y = f (x ) = -3x
2
+ 6x - 4
here a = -3, b = 6 and c = -4
(1) opens downwards since a < 0
(2) y –intercept: -4
(3) x -intercepts from -3x 2 + 6x - 4 = 0
(there are no x -intercepts here)
(4) vertex at (1, -1)
- b - b 

,f 
 
 2a  2a  
Vertex is below x-axis, and parabola opens down!
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y
1
(1, -1)
-1
y = -3x
-4
x
2
2
+ 6x - 4
(0, -4)
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The Quadratic Formula
It is not always easy to find x -intercepts
by factoring ax 2 + bx + c when solving
ax
2
+ bx + c = 0
Quadratic equations of this form can be
solved for x using the formula:
x
- b  b  4ac
2
2a
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Example: Solve x
2
− 6x + 9 = 0
here a = 1, b = -6 and c = 9
- b  b  4ac
x
2a
2
Note: the expression
inside the radical is called
the “discriminant”
- (-6)  (-6)  4(1)(9)
x
2(1)
2
6 0
Note: discriminant = 0
x 
one solution
2
6

2
 3 only as found previously
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Example: Solve x
2
-x-2=0
here a = 1, b = -1 and c = -2
x 
 ( 1) 
( 1)  4(1)( 2)
2(1)
2
1 9 13
x

2
2
1 3
13
x
or x 
2
2
x  1 or x  2
Note: discriminant > 0
two solutions
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Example: Find x -intercepts of y = x
2
-9
Solve x 2 - 9 = 0
a = 1, b = 0, c = -9
x 
0
x 

 4 ( 9 )
2
36
6

2
2
Note: discriminant > 0
two solutions
x = 3 or x = -3
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Example: Find the x -intercepts of
y = f (x) = -3x
Solve -3x
x 
x 
b
6
2
2
+ 6x - 4
a = -3, b = 6 and c = -4
+ 6x - 4 = 0
b2  4ac
2a
36  4( 3)( 4 )
2( 3)
 6   12
x 
 ?
6
Note: discriminant < 0
no Real solutions
 12 is undefined
 there are no x -intercepts as we discovered in
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an earlier plot of y = -3x 2 + 6x - 4
The end.
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