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Power Functions and Polynomial Functions: A Comprehensive Guide - Prof. Raya Ojeda, Study notes of Mathematics

A comprehensive guide to power functions and polynomial functions, covering their definitions, properties, and applications. It explores the end behavior of power functions, the degree and leading term of polynomials, and the relationship between the degree and the number of x-intercepts and turning points. The document also includes examples and exercises to reinforce understanding.

Typology: Study notes

2023/2024

Uploaded on 12/11/2024

miabella-correa
miabella-correa 🇺🇸

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3.3 POWER FUNCTIONS AND POLYNOMIAL FUNCTIONS
A power function is a function that can be represented in the form
f (x) = kxp where k and p are real numbers, and k is known as the
coefficient.
Ex. Is f (x) = 2x a power function?
No. A power function contains a variable base raised to a fixed power.
This function has a constant base raised to a variable power. This is
called an exponential function, not a power function.
EX. Which of the following functions are power functions?
f (x) = 1
f (x) = x
f (x) = x2
f (x) = x3
f (x) = 1
𝑥
f (x) = 1
𝑥2
f (x) = 𝑥
f (x) = 𝑥
3
pf3
pf4
pf5
pf8
pf9
pfa

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3.3 POWER FUNCTIONS AND POLYNOMIAL FUNCTIONS

A power function is a function that can be represented in the form f (x) = kx p where k and p are real numbers, and k is known as the coefficient. Ex. Is f (x) = 2 x a power function? No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function. EX. Which of the following functions are power functions? f (x) = 1 f (x) = x f (x) = x^2 f (x) = x^3 f (x) = 1 𝑥 f (x) = 1 𝑥^2 f (x) = (^) √𝑥 f (x) = (^) √𝑥 3

Identifying End Behavior of Power Functions As the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.

End behavior of power functions in the form f (x) = kx n where n is a non-negative integer depending on the power and the constant.

EX. Graph f(x) = x 8 and g(x) = - x 9 using a graphing calculator. EX. Which of the following are polynomial functions? a) f (x) = 2x^3 ⋅ 3x + 4 b) g(x) = −x(x 2 − 4) c) h(x) = 5 (^) √𝑥 + 2

EX. Given the function f (x) = −3x 2 (x − 1)(x + 4), express the function as a polynomial in general form, and determine the leading term, degree.

Try It # Without graphing the function, determine the maximum number of x- intercepts and turning points for f (x) = 108 − 13x^9 − 8x^4 + 14x^12 + 2x^3