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Quadratic Functions
A quadratic function is a function of the form
where are constants, and
.
The graph of a quadratic function is called a
• When a > 0,
• When a < 0,
The turning point on the parabola is called the
The vertical line passing through the vertex is called the
Example. Graph the function using translation of y
= x^{2}. Find
the vertex , axis of symmetry, and intercepts .
y = x^{2} − 6x + 8
The Graph of y = ax^{2} Let
g(x) = x^{2}, h(x) = 2x^{2},
and j(x) = −2x^{2}.
To summarize , y = ax^{2} is a parabola, similar to y = x^{2},
and
• If a < 0, then the graph opens
• If a > 0, then the graph opens
• If a > 1, then the graph opens
• If a < 1, then the graph opens
Extreme Values
A quadratic function will have a
when
A quadratic function will have a
when
This will always happen at the
Find the maximum / minimum output for the following functions:
• f(x) = x^{2} − 4x + 3
• f(x) = −2x^{2} + 6x − 9
• f(x) = 4x^{2} + 8x + 3
The Vertex Form of a Quadratic Function
The equation of the parabola y = ax^{2}+bx+c can always be rewritten
as
where the is
and the
is
Example. Find the quadratic function which passes
through the
point (−2, 3) and has a vertex of (1, 5).
Example. For what value of c will the minimum value
of f(x) =
x^{2} − 4x + c be 7?
Example. For what value of c will the maximum value
of f(x) =
x^{2} + 6x + c be 12?
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