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## Quadratic Polynomials and Functions (Definition)

We assume that you are familiar with expanding the product of two linear functions:

(x + a)(x + b)

A quadratic polynomial in x is an expression of the form ax2 + bx+ c, where a 0.

Here are some examples:

x2x − 2

x2 − 1

2x2 + x + 1

(x − 2)(x − 3) = x2 − 5x + 6

A quadratic function is a function of the form y = ax2 + bx+ c, where a 0.

We first look at functions of the form y = ax2.
Since x2 ≥ 0 for any x, the value of y = ax2 has the same sign as a.

 a y = x2 x = -2 x = -1 x = 0 x = 1 x = 2 -2 y = -2x2 -8 -2 0 -2 -8 -1 y = -x2 -4 -1 0 -1 -4 1 y = x2 4 1 0 1 4 2 y = 2x2 8 2 0 2 8

Note that at each x-value, the magnitude of y = ax2 grows as the magnitude of a grows.

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