Answer to Question #274983 in Algebra for david

Question

Reflect on the concept of polynomial and rational functions. What concepts (only the names) did you need to accommodate these concepts in your mind? What are the simplest polynomial and rational function you can imagine? In your day to day, is there any occurring fact that can be interpreted as polynomial and rational functions? What strategy are you using to get the graph of polynomial and rational functions?

Answer 1

A function is an equation where for any $x$ in the domain of the equation, the equation will yield exactly one value of $f(x)$ when the equation is evaluated at a specific $x$

The process of accomodating or adaptation of new ideas or information or new experience with polynomial and rational function is based on conceptualisation of Rational functions like fractions, terminating decimals, recurring decimals, positive and negative integers.

To assimilate new knowledge of polynomial functions, names like algebraic expressions, Quadratic expressions, cubic functions, linear equations, allow better perception.

A rational function is generalized as a ratio of two polynomial where the denominator is not equal to zero.

$f(x)=\frac{g(x)}{h(x)}$ Where $h(x)\neq0$

The names that defined polynomial functions can accommodate the definition as sum of finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power.

It is represented as

$a_nx^n+a_{n-1}x^{n-1}......a_1x+a_0$

Where all power are non-negative

and $a_0,a_1,a_2......\in\R$

From the definition the smallest Rational function will be a constant integer. eg 1

The smallest polynomial function will be of degree zero, or constant function.

Speed - distance time relationship is an application of rational function in real life while polynomial functions are used in economics to do cost analysis.

To sketch a rational function graph, data about the asymptotes , if any is crucial. With knowledge of y- and x- intercepts, a rough sketch of rational function can be obtained.

Graph of $P(x)$ depends upon its degree .

Degree zero and 1 are linear graphs and they can be easily drawn.

For sketch of degree 2 and higher, data regarding

a). x- and y- intercept

b) Stationary points and their nature

Is necessary to determine rough shape of the graphs.