Question

How do you graph a polynomial function?

Answer 1

f(x) is said to be a polynomial function if the function only depends on a single variable (x) and can be written as a sum of different multiples of x raised to some whole number. The basic step is to find the x and y intercepts.

Complete step by step solution:
Let us first understand what a polynomial function is.
Suppose we have a function $f(x)$, where f is a function of x.
Here, f(x) is said to be a polynomial function if the function only depends on a single variable (x) and can be written as a sum of different multiples of x raised to some whole number. Each term of the expression can have a real coefficient.
In other words, the polynomial functions as each term in the form of $a{{x}^{n}}$, where ‘a’ a real coefficient and n is any whole number.
The highest power of x in the function is called the degree of the polynomial.
Therefore, a nth degree polynomial function is given as $f(x)={{a}_{0}}{{x}^{0}}+{{a}_{1}}{{x}^{1}}+{{a}_{2}}{{x}^{2}}+{{a}_{3}}{{x}^{3}}+.....+{{a}_{n}}{{x}^{n}}$ …. (i)
Now, to graph a polynomial function, we need some known points.
These points can be the x and y intercepts of the function.
We can find the x intercepts by equating the function f(x) to zero.
i.e. $f(x)=0$
Then, we can find the y intercept by substituting $x=0$ in equation (i).
From the concept of continuity, we will get that a polynomial function is always continuous and form the concept of differentiation, we will get that it is differentiable at every point. Therefore, a polynomial curve is continuous and smooth.
Then, we can substitute some values of x in the function and find the value of f(x) for those values of x.
We can differentiate the function and equate it to zero to find the points where the function takes a U-turn or has a maxima or minima.
These steps are enough to plot the graph of a polynomial function.

Note:
The step of differentiating the polynomial function and equating it to zero is an essential step if the polynomial is of higher degree as it helps us to predict how the function behaves in a specific interval (where it takes a turn).
The number of U turns in a polynomial curve can be (n-1), at the most.