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Chapter 2: Problem 78

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

Short Answer

Expert verified
The graph of a polynomial function is described as 'smooth' meaning it has no sharp corners or turns and 'continuous' meaning it has no holes, jumps, or vertical asymptotes. In simple terms, you can draw it without lifting your pen.

Step by step solution

01

Understand the term 'smooth' in relations to a graph

The term 'smooth' when describing a graph, particularly a graph of a polynomial function, means that the graph has no sharp turns or corners. This is because all of the derivatives of a polynomial function exist, which makes the graph smooth everywhere.
02

Understand the term 'continuous' in relation to a graph

When a graph is described as 'continuous', it implies that the graph has no holes, jumps, or vertical asymptotes. In other words, you can draw the whole graph without lifting your pen.
03

Relating these terms to a polynomial function

Since all polynomials are smooth and continuous, the graph of any polynomial function will be a smooth, continuous curve with no sharp corners, holes, or vertical asymptotes.

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Most popular questions from this chapter

Is every rational function a polynomial function? Why or why not? Does a true statement result if the two adjectives rational and polynomial are reversed? Explain.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When solving \(f(x)>0,\) where \(f\) is a polynomial function, I only pay attention to the sign of \(f\) at each test value and not the actual function value.

What is a rational inequality?

A company that manufactures running shoes has a fixed monthly cost of \(\$ 300,000\). It costs \(\$ 30\) to produce each pair of shoes. a. Write the cost function, \(C,\) of producing \(x\) pairs of shoes. b. Write the average cost function, \(\bar{C},\) of producing \(x\) pairs of shoes. c. Find and interpret \(\bar{C}(1000), \bar{C}(10,000),\) and \(\bar{C}(100,000)\) d. What is the horizontal asymptote for the graph of the average cost function, \(\bar{C} ?\) Describe what this represents for the company.

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{x^{2}}-3$$
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