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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 = x2. Find
the vertex , axis of symmetry, and intercepts .

y = x2 − 6x + 8

The Graph of y = ax2 Let g(x) = x2, h(x) = 2x2,
and j(x) = −2x2.

To summarize , y = ax2 is a parabola, similar to y = x2, 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

Graph the following :

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) = x2 − 4x + 3

• f(x) = −2x2 + 6x − 9

• f(x) = 4x2 + 8x + 3

The Vertex Form of a Quadratic Function

The equation of the parabola y = ax2+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) =
x2 − 4x + c be -7?

Example. For what value of c will the maximum value of f(x) =
x2 + 6x + c be 12?

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