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Chapter 3: Problem 78

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.

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Most popular questions from this chapter

If \(f(x)=-x^{3}+4 x,\) then the graph of \(f\) falls to the left and falls to the right.

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. It is possible to have a rational function whose graph has no \(y\) -intercept.

Describe a strategy for graphing a polynomial function. In your description, mention intercepts, the polynomial鈥檚 degree, and turning points.

Use long division to rewrite the equation for \(g\) in the form $$ \text {quotient}+\frac{\text {remainder}}{\text {divisor}} $$ Then use this form of the function's equation and transformations \( \text { of } f(x)=\frac{1}{x} \text { to graph } g \). $$ g(x)=\frac{3 x-7}{x-2} $$

What are the zeros of a polynomial function and how are they found?
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