Question

Which of our toolkit functions are power functions? The constant and identity functions are power functions, since they can be written as f(x) = x° and f(x) = x' respectively. This chapter is part of Precalculus: An Investigation of Functions Lippman & Rasmussen 2017, and contains content remixed with permission from College Algebra C Stitz & Zeager 2013. This material is licensed under a Creative Commons CC-BY-SA license. 160 Chapter 3 | The quadratic and cubic functions are both power functions with whole number powers: f(x) = x’ and f(x)=x?. The reciprocal and reciprocal squared functions are both power functions with negative whole number powers since they can be written as f(x) = x-hand f(x)=x². The square and cube root functions are both power functions with fractional powers since they can be written as f(x) = x2 or f(x)=xV. Example 2 Describe the long run behavior of the graph of f(x) = x®. Since f(x)= x® has a whole, even power, we would expect this function to behave somewhat like the quadratic function. As the input gets large positive or negative, we would expect the output to grow without bound in the positive direction. In symbolic form, as x, f(x) →. Example 3 Describe the long run behavior of the graph of f(x) = -x' Since this function has a whole odd power, we would expect it to behave somewhat like the cubic function. The negative in front of the x' will cause a vertical reflection, so as the inputs grow large positive, the outputs will grow large in the negative direction, and as the inputs grow large negative, the outputs will grow large in the positive direction. In symbolic form, for the long run behavior we would write: as x, f(x) - and as x--, f(x) +0. O Even numbered exponents are power functions that look like a parabola, while odd numbered exponents look like a cubic function. Use the arrow notation introduces in Examples 1 - 3 to answer the following: (a) What is the long run behavior for an even power function with a negative leading coefficient? (b) What is the long run behavior for an odd power function with a positive leading coefficient? Give an example of a 5th degree polynomial with a leading coefficient of 4 that has at least two terms. * What information is the textbook asking for when they say they want you to describe the short term behavior of a polynomial?

Answer 1

An even power function with a negative leading coefficient has a long run behavior of negative infinity, while an odd power function with a positive leading coefficient has a long run behavior of positive infinity. An example of a 5th degree polynomial with a leading coefficient of 4 that has at least two terms is 4x^5 + 2x^2.

Step-by-step explanation:

An even power function with a negative leading coefficient will have a long run behavior where the output grows without bound in the negative direction as the input grows large positive or negative. In symbolic form, as x → −∞, f(x) → −∞.

An odd power function with a positive leading coefficient will have a long run behavior where the output grows without bound in the positive direction as the input grows large positive or negative. In symbolic form, as x → ∞, f(x) → ∞.

An example of a 5th degree polynomial with a leading coefficient of 4 that has at least two terms can be expressed as f(x) = 4x^5 + 2x^2.