Sketching quadratic functions

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Sketching quadratic functions
To sketch a quadratic function we need to
identify where possible
The y intercept (0, c)
The roots by solving ax2 bx c 0
The axis of symmetry (mid way between the roots)
The coordinates of the turning point.
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The shape
The coefficient of x2 is -1 so the shape is
The Y intercept
(0 , 5)
The roots
(-5 , 0) (1 , 0)
The axis of symmetry
Mid way between -5 and 1 is -2
x -2
The coordinates of the turning point
(-2 , 9)
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Completing the square
The coordinates of the turning point of a
quadratic can also be found by completing the
square.
This is particularly useful for parabolas that do
not cut the x axis.
REMEMBER
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Axis of symmetry is x 2
Coordinates of the minimum turning point is (2 ,
1)
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Axis of symmetry is x 3
Coordinates of the maximum turning point is (3 ,
16)
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Solving quadratic equations
Quadratic equations may be solved by
The Graph
Factorising
Completing the square
Using the quadratic formula
7
This does not factorise.
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Quadratic inequations
A quadratic inequation can be solved by using a
sketch of the quadratic function.
First do a quick sketch of the graph of the
function.
Roots are -4 and 1.5
The function is positive when it is above the x
axis.
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First do a quick sketch of the graph of the
function.
Roots are -4 and 1.5
The function is negative when it is below the x
axis.
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The quadratic formula
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From the above example when the number under the
square root sign is zero there is only 1 solution.
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From the above example we require the number
under the square root sign to be positive in
order for 2 real roots to exist.
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This leads to the following observation.
Since the discriminant is zero, the roots are
real and equal.
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Using the discriminant
We can use the discriminant to find unknown
coefficients in a quadratic equation.
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Since the discriminant is always greater than or
equal to zero, the roots of the equation are
always real.
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Conditions for tangency
To determine whether a straight line cuts,
touches or does not meet a curve the equation of
the line is substituted into the equation of the
curve.
When a quadratic equation results, the
discriminant can be used to find the number of
points of intersection.
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Since the discriminant is zero, the line is a
tangent to the curve.
Hence the point of intersection is (1 , 1).
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Hence the equation of the tangent is y 2x.
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Hence the equation of the two tangents are y 8x
2 and y -8x - 2.