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.
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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.
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
How does the graph of a quadratic function change as we change a, b, and c?
Parabolas
a decreases from 1 towards 0
How does the graph of a quadratic function change as we change a, b, and c?
Parabolas
a increases from 1 to 10
How does the graph of a quadratic function change as we change a, b, and c?
Parabolas
c increases from 0 to 2
How does the graph of a quadratic function change as we change a, b, and c?
Parabolas
c decreases from 0 to -2
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
SummaryGeneral Form:
Vertex:
Axis of symmetry:
standard Form:
Vertex:
Axis of symmetry:
Parabola opens up
Parabola opens down
problemsFind the vertex and the x-intercepts of the following functions:
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)
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?