# 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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