Section 2.7 Graphs of Functions 205 Linear Functions and Constant Functions Let m and b represent real numbers such that m 0. Then A function that can be written in the form f1x2 mx b is a linear function. A function that can be written in the form f1x2 b is a constant function. Note: The graphs of linear and constant functions are lines. 2. Graphs of Basic Functions At this point, we are able to recognize the equations and graphs of linear and constant functions. In addition to linear and constant functions, the following equations define six basic functions that will be encountered in the study of algebra: Equation Function Notation Type of Function Identity function Quadratic function Cubic function Absolute value function Square root function f 1x2 y x x f 1x2 y x2 x2 f 1x2 y x3 x3 f 1x2 y 0x 0 0x 0 f 1x2 y 1x 1x 1 x f Reciprocal function 1x2 y 1 x equivalent function notation f1x2 m 1 x The graph of the function defined by is linear, with slope and y-intercept (0, 0) (Figure 2-37). To determine the shapes of the other basic functions, we can plot several points to establish the pattern of the graph. Analyzing the equation itself may also provide insight f1x2 to the domain, range, and shape of the graph. To demonstrate this, we will graph x2 and . g1x2 1x Graphing Basic Functions f1x2 x2 Example 1 Graph the function defined by . Solution: The domain of the function given by f1x2 x2 1 or equivalently y x22 is all real numbers. To graph the function, choose arbitrary values of x within the domain of the function. Be sure to choose values of x that are positive and values that are negative to determine the behavior of the function to the right and left of the origin (Table 2-6).The graph of is shown in Figure 2-38. f1x2 f1x2 x2 The function values are equated to the square of x, so will always be greater than or equal to zero. Hence, the y-coordinates on the graph will never 4 3 2 5 4321 1 2 3 4 5 1 5 4 y 1 54 3 1 2 3 4 5 21 1 2 3 4 5 Figure 2-37 x 3 2 f(x) x x f (x) f(x) 2x 3 5 1 2 3 4 5 Figure 2-36 An equation of the form y mx b is represented graphically by a line with slope m and y-intercept (0, b). In function notation, this may be written as f(x) mx b. A function in this form is called a linear function. For example, the function defined by f(x) 2x 3 is a linear function with slope m 2 and y-intercept (0, 3) (Figure 2-36). ⎫⎪⎪⎪⎬⎪⎪⎪⎭
miller_intermediate_algebra_4e_ch1_3
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