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Quadratic Functions

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A polynomial functionis a function that can be expressed in the form:

p(x)=an xn + an-1 xn-1 + … + a1x + a0

Where an, an-1 , … , a1, a0are real numbers, an≠ 0, the exponents are non-negative integers

Definition:

✔

The degree is 2

✔

The degree is 1

✔

The degree is 0

✔

The degree is 3

The degree of a polynomial is largest exponent of x.

Definition:

A polynomial of degree 0 is called a constant function.

A polynomial of degree 1 is called a linear function.

A degree 2polynomial function is called a quadratic function. The general form a quadratic function is

p(x) = ax2 + bx + c

where a, b, and c are real numbers with a ≠ 0.

Definition:

Quadratic functions are incredibly important functions that show up everywhere in the real world.

The graph of a quadratic polynomial is called a parabola.

p(x)= ax2 + bx + c

Axis of

Symmetry

vertex

Axis of

Symmetry

vertex

a > 0

a < 0

How does the graph of a quadratic function change as we change a, b, andc?

adecreases from 1 towards 0

How does the graph of a quadratic function change as we change a, b, andc?

a increases from 1 to 10

How does the graph of a quadratic function change as we change a, b, andc?

c increases from 0 to 2

How does the graph of a quadratic function change as we change a, b, andc?

c decreases from 0 to -2

The standard form of a quadratic function is

p(x) = a(x – h)2 + k

Where (h, k) is the vertex of its graph and a ≠ 0.

Definition:

General Form:

standard Form:

Vertex:

Vertex:

Axis of symmetry:

Axis of symmetry:

Parabola opens up

Parabola opens down

Find the vertex and the x-intercepts of the following functions:

Find the quadratic functionwith the indicated vertex that passing though the given point:

1. Vertex: (2,3)

Point: (0,2)

2. Vertex: (-2,-2)

Point: (-1,0)

3. Vertex: (6,6)

Point: (1/2, 3/4)

The profit P (in hundreds of dollars) that a company makes depends on the amount x (in hundreds of dollars) that the company spends on advertising according to the model:

P(x) = 230 + 20x – 0.5x2

How much should the company spend on advertising to maximize profits?