Graphing Functions of the Form x^n

As we probably already know, a polynomial function is a function in which f(x) is a collection of terms of x raised to whole number powers. But what happens when we have polynomials with varying degrees? How do we graph these functions? Let''s find out:

Graphing polynomial functions with degrees

Consider the following function:

f(x) = a_nxⁿ+a_n-1x^n-1+a_n-2x^n-2+..+a₂x²+a₁x+a₀

This is an example of a polynomial function with the degree "n."

We know that when we have a polynomial function with a degree of zero such as:

f(x) = a*x^0 = a

we are left with a horizontal line:

But what happens if we have a polynomial function of degree 1? This function takes the following form:

f(x) = ax+b

, where a cannot equal zero. Here''s what a graph of this function looks like:

What about a polynomial function with degree 2? This is called a quadratic equation, and it takes the following form:

f(x) = ax²+bx+c

where a cannot equal zero. As we will see, this function takes the form of a parabola when graphed:

When we get into the higher degrees, things can get a little more complicated. One important note is that the form of these graphed functions changes depending on whether the exponent is even or odd. Take a look:

If we get stuck, we might need to factor the polynomials to find the x-intercepts. Calculus can also be very handy when determining the critical points of a polynomial function.

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