Problem 57 - College Algebra

Question

What do we mean when we describe the graph of a polynomial function as smooth and continuous?

Step-by-Step Solution

Answer

When we describe the graph of a polynomial function as smooth and continuous, we mean that the graph has no breaks, holes or sharp corners (implying it is smooth) and you can draw it without lifting your pen from the paper (implying it is continuous). This means polynomial functions have derivatives that exist at every point in their domain.

Step 1: Defining a Polynomial Function

A polynomial function is a function that can be expressed in the form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\), where \(a_i\) are constants for \(i = 0,1, ..., n\) and \(n\) is a nonnegative integer.

Step 2: Defining Smoothness

In mathematics, a function is described as smooth if it has derivatives of all orders everywhere in its domain. In other words, the function is smooth if it does not have any jumps or kinks, or any sudden changes in direction. All polynomial functions are smooth functions.

Step 3: Defining Continuity

A function is said to be continuous if it does not have any holes, jumps or vertical asymptotes. Put simply, if a function is continuous, you could draw it without lifting your pen from the paper. All polynomial functions are continuous everywhere. This is because for each value of \(x\), there exists a corresponding value of \(y=f(x)\).

Step 4: Conclusion

So, when we describe the graph of a polynomial function as smooth and continuous, we mean that its graph has no breaks, holes, jumps, or kinks and that it can be drawn without lifting the pen from the paper, signifying that it has derivatives of all orders everywhere in its domain.