Problem 78 - Algebra and Trigonometry

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, we mean it has no sharp corners or sudden changes in direction. When we say it is continuous, we mean that it has no breaks, jumps, or holes; there is a defined value in the range for every real number in the domain.

Step 1: Define 'Smooth'

In mathematics, when we describe a graph as 'smooth', this means that the graph doesn't have any sharp 'corners' or 'jagged edges'. In terms of a polynomial function, it means that the graph can be drawn without lifting the pencil from the paper, and still maintaining a continuous line which curves smoothly without sudden changes of direction.

Step 2: Define 'Continuous'

When we describe a graph of a function as 'continuous', we are stating that it has no breaks, jumps, or holes. For all real numbers within the domain of the function, there will be a corresponding value in the range. For polynomial functions, this is always the case, as polynomials are defined for all real values of x.