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Chapter 1 | Functions and Graphs
1.2 | Basic Classes of Functions
Learning Objectives 1.2.1 Calculate the slope of a linear function and interpret its meaning.
1.2.2 Recognize the degree of a polynomial. 1.2.3 Find the roots of a quadratic polynomial. 1.2.4 Describe the graphs of basic odd and even polynomial functions. 1.2.5 Identify a rational function. 1.2.6 Describe the graphs of power and root functions. 1.2.7 Explain the difference between algebraic and transcendental functions. 1.2.8 Graph a piecewise-defined function. 1.2.9 Sketch the graph of a function that has been shifted, stretched, or reflected from its initial graph position. We have studied the general characteristics of functions, so now let’s examine some specific classes of functions. We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. 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 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. Linear Functions and Slope The easiest type of function to consider is a linear function . Linear functions have the form f ( x ) = ax + b , where a and b are constants. In Figure 1.15 , we see examples of linear functions when a is positive, negative, and zero. Note that if a >0, the graph of the line rises as x increases. In other words, f ( x ) = ax + b is increasing on (−∞, ∞). If a <0, the graph of the line falls as x increases. In this case, f ( x ) = ax + b is decreasing on (−∞, ∞). If a =0, the line is horizontal.
Figure 1.15 These linear functions are increasing or decreasing on (∞, ∞) and one function is a horizontal line.
As suggested by Figure 1.15 , the graph of any linear function is a line. One of the distinguishing features of a line is its slope. The slope is the change in y for each unit change in x . The slope measures both the steepness and the direction of a line. If the slope is positive, the line points upward when moving from left to right. If the slope is negative, the line points downward when moving from left to right. If the slope is zero, the line is horizontal. To calculate the slope of a line, we need to determine the ratio of the change in y versus the change in x . To do so, we choose any two points ( x 1 , y 1 ) and ( x 2 , y 2 ) on the line and calculate . In Figure 1.16 , we see this ratio is independent of the points chosen.
y 2 − y 1 x 2 − x 1
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