quadratic functions. polynomials definition : a polynomial function is a function that can be...

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QuadraticFunctions

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

p(x) = an xn + an-1 xn-1 + … + a1 x + a0

Where an , an-1 , … , a1 , a0 are real numbers, an ≠ 0, the exponents are non-negative integers

Definition: The degree of a polynomial is largest exponent of x.

✔✔

The degree is 2

The degree is 1

The degree is 0

The degree is 3

Quadraticfunctions

A polynomial of degree 0 is called a constant function.

A polynomial of degree 1 is called a linear function.

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

p(x) = ax2 + bx + cwhere a, b, and c are real numbers with a ≠ 0.

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

Page 4: Quadratic Functions. PolyNomials Definition : A polynomial function is a function that can be expressed in the form: p(x) = a n x n + a n-1 x n-1 + …

The graph of a quadratic polynomial is called a parabola.

Parabolas

p(x) = ax2 + bx + c

a > 0 a < 0

vertex

vertexAxis of Symmetry

Axis of Symmetry

Page 5: Quadratic Functions. PolyNomials Definition : A polynomial function is a function that can be expressed in the form: p(x) = a n x n + a n-1 x n-1 + …

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

Parabolas

a decreases from 1 towards 0

Page 6: Quadratic Functions. PolyNomials Definition : A polynomial function is a function that can be expressed in the form: p(x) = a n x n + a n-1 x n-1 + …

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

Parabolas

a increases from 1 to 10

Page 7: Quadratic Functions. PolyNomials Definition : A polynomial function is a function that can be expressed in the form: p(x) = a n x n + a n-1 x n-1 + …

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

Parabolas

c increases from 0 to 2

Page 8: Quadratic Functions. PolyNomials Definition : A polynomial function is a function that can be expressed in the form: p(x) = a n x n + a n-1 x n-1 + …

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

Parabolas

c decreases from 0 to -2

Page 9: Quadratic Functions. PolyNomials Definition : A polynomial function is a function that can be expressed in the form: p(x) = a n x n + a n-1 x n-1 + …

Definition: The standard form of a quadratic function is

p(x) = a(x – h)2 + kWhere (h, k) is the vertex of its graph and a ≠ 0.

Standardform

Page 10: Quadratic Functions. PolyNomials Definition : A polynomial function is a function that can be expressed in the form: p(x) = a n x n + a n-1 x n-1 + …

SummaryGeneral Form:

Vertex:

Axis of symmetry:

standard Form:

Vertex:

Axis of symmetry:

Parabola opens up

Parabola opens down

Page 11: Quadratic Functions. PolyNomials Definition : A polynomial function is a function that can be expressed in the form: p(x) = a n x n + a n-1 x n-1 + …

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

Page 12: Quadratic Functions. PolyNomials Definition : A polynomial function is a function that can be expressed in the form: p(x) = a n x n + a n-1 x n-1 + …

problemsFind the quadratic function with 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)

Page 13: Quadratic Functions. PolyNomials Definition : A polynomial function is a function that can be expressed in the form: p(x) = a n x n + a n-1 x n-1 + …

problemsThe 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?

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