2.4. A LIBRARY OF PARENT FUNCTIONS

Transcription

1 2.4. A LIBRARY OF PARENT FUNCTIONS 1

2 What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal function. Identify and graph step and other piecewise-defined functions. Recognize graphs of parent functions. 2

3 Linear Functions One of the goals of this text is to enable you to recognize the basic shapes of the graphs of different types of functions. For instance, you know that the graph of the linear function f (x) = ax + b is a line with slope m = a and y-intercept at (0, b). 3

4 Linear Functions The graph of the linear function has the following characteristics. The domain of the function is the set of all real numbers. The range of the function is the set of all real numbers. The graph has an x-intercept of ( b/m, 0) and a y-intercept of (0, b). The graph is increasing if m 0, decreasing if m 0, and constant if m = 0. 4

5 Example Writing a Linear Function Write the linear function f for which f (1) = 3 and f (4) = 0. 5

6 Solution To find the equation of the line that passes through (x 1, y 1 ) = (1, 3) and (x 2, y 2 ) = (4, 0) first find the slope of the line. 6

7 Solution Next, use the point-slope form of the equation of a line. y y 1 = m(x x 1 ) y 3 = 1(x 1) y = x + 4 f (x) = x + 4 Point-slope form Substitute for x 1, y 1 and m Simplify. Function notation 7

8 Solution The graph of this function is shown in Figure Figure

9 Constant Function There are two special types of linear functions, the constant function and the identity function. A constant function has the form f (x) = c and has the domain of all real numbers with a range consisting of a single real number c. The graph of a constant function is a horizontal line. 9

10 Identity Functions The identity function has the form f (x) = x. Its domain and range are the set of all real numbers. The identity function has a slope of m = 1 and a y-intercept at (0, 0). The graph of the identity function is a line for which each x-coordinate equals the corresponding y-coordinate. The graph is always increasing. 10

11 Squaring Functions The graph of the squaring function f (x) = x2 is a U-shaped curve with the following characteristics. The domain of the function is the set of all real numbers. The range of the function is the set of all nonnegative real numbers. The function is even. The graph has an intercept at (0, 0). 11

12 Squaring Functions The graph is decreasing on the interval ( increasing on the interval (0, ), 0) and The graph is symmetric with respect to the y-axis. The graph has a relative minimum at (0, 0). 12

13 Cubic, Square Root, and Reciprocal Functions The basic characteristics of the graphs of the cubic, square root, and reciprocal functions are summarized below. 1. The graph of the cubic function f (x) = x 3 has the following characteristics. The domain of the function is the set of all real numbers. The range of the function is the set of all real numbers. The function is odd. The graph has an intercept at (0, 0). 13

14 Cubic Functions The graph is increasing on the interval (, ). The graph is symmetric with respect to the origin. Cubic function 14

15 Square Root Functions 2. The graph of the square root function f (x) = has the following characteristics. The domain of the function is the set of all nonnegative real numbers. The range of the function is the set of all nonnegative real numbers. The graph has an intercept at (0, 0). The graph is increasing on the interval (0, ) 15

16 The graph of the Square Root Functions Square root function 16

17 Reciprocal Functions 3. The graph of the reciprocal function f (x) = has the following characteristics. The domain of the function is(, 0) (0, ) The range of the function is (, 0) (0, ) The function is odd. The graph does not have any intercepts. The graph is decreasing on the intervals (, 0) and (0, ). The graph is symmetric with respect to the origin. 17

18 Reciprocal Functions Reciprocal function 18

19 Step Functions Functions whose graphs resemble sets of stairsteps are known as step functions. The most famous of the step functions is the greatest integer function, which is denoted by defined as and f (x) = = the greatest integer less than or equal to x. Some values of the greatest integer function are as follows. = (greatest integer 1) = 1 = (greatest integer ) = 1 19

20 Step Functions = (greatest integer ) = 0 = (greatest integer 1.5) = 1 The graph of the greatest integer function f (x) = has the following characteristics: The domain of the function is the set of all real numbers. The range of the function is the set of all integers. The graph has a y-intercept at (0, 0) and x-intercepts in the interval [0, 1). 20

21 Step Functions The graph is constant between each pair of consecutive integers. The graph jumps vertically one unit at each integer value. 21

22 Example Evaluating a Step Function Evaluate the function when x = 1, 2 and f (x) =

23 Solution For x = 1, the greatest integer 1 is 1, so f ( 1) = + 1 = = 0 23

24 Solution For x = 2, the greatest integer 2 is 2, so f (2) = + 1 = = 3. For x =, the greatest integer is 1, so = = 2 24

25 Solution You can verify your answers by examining the graph of f (x) =

26 Parent Functions The eight graphs shown in Figure 2.48 represent the most commonly used functions in algebra. (a) Constant Function (b) Identity Function (c) Absolute Value Function (d) Square Root Function Figure

27 Parent Functions cont d (e) Quadratic Function (f) Cubic Function Figure 2.48 (g) Reciprocal Function (h) Greatest Integer Function Familiarity with the basic characteristics of these simple graphs will help you analyze the shapes of more complicated graphs in particular, graphs obtained from these graphs by the rigid and non-rigid transformations. 27