UNIT 2: POLYNOMIAL FUNCTIONS AND RATIONALS

Definition(meaning what is?): A polynomial function is a function that can be written in the form f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+…a1x+a0

where n is a positive integer and an to a. are real numbers with an not being equal to zero.

Definition( meaning of what is) The degree of a polynomial is the highest exponent of all the terms.

Degree 1 (Linear Function ): A polynomial of degree 1 is known as a linear function. It can be written in the form y=mx+b

Degree 2 (Quadratic Function): A polynomial of degree 2 is known as a quadratic function. The quadratic function can be written in different forms. Below are the forms of a quadratic function

Standard form y=ax^2+bx+c (Best form for finding the y-intercept=c)

Vertex form y=a[b(x-h)]^2+k (Best form for finding the vertex and consequently the shifts.

Vertex form y=a[b(x-h)]^2+k (Best form for finding the vertex and consequently the shifts.

Degree 3(Cubic Functions) A polynomial of degree 3 is know as a cubic function

A Quadratic function is a polynomial of degree 2.

Given a quadratic function with vertex form y=a[b(x-h)]+k, and standard form y=ax2+bx+c
The properties are as follows

Vertex (h,k)

Opening: Opens up if a>0, and opens down if a

Extrema: Maximum if a>0, minimum if a<0. Extrema at h=-b/2a, and value at k = f(-b/2a)

Shift: h units left for f(x +h) and h units right for f(x-h), k units up for f(x)+k, and k units for f(x)-k

Axis of symmetry: x=h or x=-b/2a

y-intercept: value of c or find f(0)

x-intercept: find x for f(x)=0

Domain: (- ꚙ,ꚙ)

Range: Opens up [h, ꚙ) Opens down (-ꚙ,h]

Polynomials of higher degrees can be generated when you are provided with sufficient information concerning the zeros.