Quadrat
ic
Functio
ns
PolyNomials
DefinitionA: 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
✔
✔
✔
The degree is 2
The degree is 1
The degree is 0
✔
The degree is 3
Definition:The degree of a polynomial is largest exponent of x.
Quadratic
functions
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 + c
where a, b, and c are real numbers with a ≠ 0.
Quadratic functions are incredibly important functions that show up
everywhere in the real world.
Parabolas
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
Parabolas
How does the graph of a quadratic function change as we change a, b, and c?
a decreases from 1 towards 0
Parabolas
How does the graph of a quadratic function change as we change a, b, and c?
a increases from 1 to 10
Parabolas
How does the graph of a quadratic function change as we change a, b, and c?
c increases from 0 to 2
Parabolas
How does the graph of a quadratic function change as we change a, b, and c?
c decreases from 0 to -2
Standard
form
Definition: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.
Summary
General
Form:
standard
Form:
Vertex:
Vertex:
Axis of symmetry:
Axis of symmetry:
Parabola opens up
Parabola opens down
problems
Find the vertex and the x-intercepts of the following functions:
problems
Find the quadratic function with the indicated vertex that passing though
the given point:
1. Vertex: (2,3)
2. Vertex: (-2,-2)
Point: (0,2)
Point: (-1,0)
3. Vertex: (6,6)
Point: (1/2, 3/4)
problems
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?