Functions

Some Elementary Functions

Polynomial Functions:

A polynomial function of degree n is a function defined by an equation of the form:

where (1) a_n, a_n-1, .., a₁, a₀ are real numbers, (2) a_n does not equal to 0, and (3) n is an integer greater or equal to 0.

The domain is . The range is when n is odd, when n is even and a_n > 0, and when n is even and a_n< 0. If P(a) = 0, a is called a real zero of P(x).

Theorem: A polynomial function P of degree n has, at most, n distinct real zeroes.

Linear Functions: Polynomial functions of degree 1 (P(x) = a₁x + a₀) are called linear functions since their graphs are straight lines with slope a₁ and y-intercept a₀ . Now see the graph of a linear function given by y = ax+b. Here you can change the values of a and b by choosing the feature you wish to change and dragging the line.

Quadratic Functions: Polynomial functions of degree 2 (P(x) = a₂x² + a₁x + a₀) are called quadratic functions. A quadratic function has at most two real zeroes, and in order to find them we have to solve the equation ax² + bx + c = 0 . The real roots are given by the quadratic formula:

,

where . Therefore, the existence of real roots depend on the value of . Why? How many real roots are there if:

Click above to see the answers!!!

A graph of a polynomial of degree 2 is a parabola concave up if a₂ > 0 and concave down if a₂ < 0 .See next, how the graph of a quadratic function (P(x)=ax²+bx+c) behave as you vary a and

Polynomial Functions of Degree Greater than 2

Cubics functions are the polynomial functions of degree 3 and are of the form P(x) = a₃x³ + a₂x² +a₁x + a₀ .

A complete analysis of graphs of polynomial functions of degree greater than 2 requires methods that you will learn in calculus. For now play with this applet which plots a graph of a polynomial function of degree at most 6. Change parameters, check the roots of P(x), reduce the degree of P(x), change ranges of x and y, and see how the graph changes.