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Chapter 3: Problem 85

Explain the relationship between the degree of a polynomial function and the number of turning points on its graph.

Short Answer

Expert verified
The maximum number of turning points a polynomial function can have is always one less than the degree of the polynomial. So, a polynomial of degree n can have up to n-1 turning points.

Step by step solution

01

Understand the basics of a polynomial function

A polynomial function is a function such as a quadratic, a cubic, a quartic, and so on, where the highest power exponent is a whole number. The highest power of the variable in a polynomial function is known as the 'degree' of the function. We'll need to understand this to proceed.
02

Turning points of a polynomial function

Turning points refer to the number of times the graph changes direction (from increasing to decreasing or vice versa). Essentially, these are the local maxima or minima. Understanding this phenomenon will help in relating the degree of polynomial with turning points.
03

Relationship between degree and turning points

The maximum number of turning points of a polynomial function could be one less than the degree of the polynomial. For example, a linear polynomial (degree 1) has 0 turning points, a quadratic polynomial (degree 2) has at most 1 turning point, a cubic polynomial (degree 3) can have at most 2 turning points, and so on.

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Most popular questions from this chapter

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degreeof \(p\) and \(q\) are as small as possible. More than one correct finction may be possible. Graph your function using a graphing utility to verify that it has the required propertics \(f\) has vertical asymptotes given by \(x=-2\) and \(x=2\) a horizontal asymptote \(y=2, y\) -intercept at \(\frac{9}{2}, x\) -intercepts at \(-3\) and \(3,\) and \(y\) -axis symmetry.

Use the four-step procedure for solving variation problems given on page 424 to solve. \(y\) varies directly as \(x . y=45\) when \(x=5 .\) Find \(y\) when \(x=13 .\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that \(f(-x)\) is used to explore the number of negative real zeros of a polynomial function, as well as to determine whether a function is even, odd, or neither.

Explain what is meant by combined variation. Give an example with your explanation.

Use the four-step procedure for solving variation problems given on page 424 to solve. \(y\) varies jointly as \(a\) and \(b\) and inversely as the square root of \(c . y=12\) when \(a=3, b=2,\) and \(c=25 .\) Find \(y\) when \(a=5, b=3,\) and \(c=9\).
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