The behavior for higher-degree polynomials can be analyzed similarly. To understand the end behavior for polynomial functions, we can focus on quadratic and cubic functions. We can describe what happens to the values of f ( x ) We can conclude that the function f ( x ) = 3 x 2Īpproaches infinity, and we write 3 x 2 → ∞Ĭan be defined similarly. In that case, we say “ f ( x )Īpproaches infinity,” and we write f ( x ) → ∞įor example, for the function f ( x ) = 3 x 2 , May not approach a finite number but instead may become larger for all values of xĪs they get larger. Is a horizontal asymptote for the function f ( x ) = 2 + 1 / xīecause the graph of the function gets closer to the line as x Goes to infinity,” and we write f ( x ) → 2 For example, for the function f ( x ) = 2 + 1 / x ,īecome closer and closer to zero for all values of xĪs they get larger and larger. For some functions, the values of f ( x )Īpproach a finite number. To determine the behavior of a function fĪs the inputs approach infinity, we look at the values f ( x )īecome larger. Is an odd function because f ( − x ) = a ( − x ) n = − a x n Is an even function because f ( − x ) = a ( − x ) n = a x n (We consider other cases later.) If the exponent is a positive integer, then f ( x ) = a x n The exponent in a power function can be any real number, but here we consider the case when the exponent is a positive integer. A power function is any function of the form f ( x ) = a x b ,Īre any real numbers. Some polynomial functions are power functions. In particular, a quadratic function has the form f ( x ) = a x 2 + b x + c , A polynomial function of degree 2 is called a quadratic function. A linear function of the form f ( x ) = m x + bĪ polynomial of degree 0 is also called a constant function. Is called the degree of the polynomial the constant a n In, we see this ratio is independent of the points chosen.į ( x ) = a n x n + a n − 1 x n − 1 + … + a 1 x + a 0 On the line and calculate y 2 − y 1 x 2 − x 1. To do so, we choose any two points ( x 1, y 1 ) To calculate the slope of a line, we need to determine the ratio of the change in y If the slope is zero, the line is horizontal. If the slope is negative, the line points downward when moving from left to right. If the slope is positive, the line points upward when moving from left to right. The slope measures both the steepness and the direction of a line. One of the distinguishing features of a line is its slope. In this case, f ( x ) = a x + bĪs suggested by, the graph of any linear function is a line. In, we see examples of linear functions when a Linear functions have the form f ( x ) = a x + b ,Īre constants. The easiest type of function to consider is a linear function. We finish the section with examples of piecewise-defined functions and take a look at how to sketch the graph of a function that has been shifted, stretched, or reflected from its initial form. By combining root functions with polynomials, we can define general algebraic functions and distinguish them from the transcendental functions we examine later in this chapter. We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. We have studied the general characteristics of functions, so now let’s examine some specific classes of functions. Sketch the graph of a function that has been shifted, stretched, or reflected from its initial graph position.Explain the difference between algebraic and transcendental functions.Describe the graphs of power and root functions.Describe the graphs of basic odd and even polynomial functions.Find the roots of a quadratic polynomial.Calculate the slope of a linear function and interpret its meaning.
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