194 CHAPTER 4 Polynomial and Rational Functions 2 Graph Polynomial Functions Using Transformations The methods of shifting, compressing, stretching, and reflecting studied in Section 2.5, when used with the facts just presented, enable us to graph polynomial functions that are transformations of power functions. Graphing a Polynomial Function Using Transformations Graph: f x x 1 2 1 4 ( ) ( ) = − EXAMPLE 3 Solution Figure 5 shows the required steps. Graphing a Polynomial Function Using Transformations Graph: f x x 1 5 ( ) = − Solution EXAMPLE 2 It is helpful to rewrite f as f x x 1. 5 ( ) = − + Figure 4 shows the required steps. Figure 4 x y 22 2 Multiply by 21; reflect about x-axis (a) y 5 x5 2 22 x y 22 2 Add 1; shift up 1 unit (b) y 5 2x5 2 22 x y 2 22 (c) y 5 2x5 1 1 5 1 2 x5 2 22 (21, 1) (21, 2) (1, 1) (0, 0) (0, 0) (0, 1) (1, 0) (21, 21) (1, 21) Check: Verify the graph of f by graphing Y x 1 1 5 = − on a graphing utility. Figure 5 x y 22 2 Replace x by x 2 1; shift right 1 unit (a) y 5 x4 2 22 x y 22 2 Multiply by ; vertical compression by a factor of (b) y 5 ( x 2 1)4 2 22 x y 2 22 2 22 (21, 1) (0, 1) (1, 1) (0, 0) (2, 1) (1, 0) 0, ( ) ( ) 2, (1, 0) 1 – 2 1 – 2 (c) y 5 ( x 2 1)4 1 – 2 1 – 2 1 – 2 Check: Verify the graph of f by graphing Y x 1 2 1 1 4 ( ) = − on a graphing utility. Now Work PROBLEMS 29 AND 35 3 Identify the Real Zeros of a Polynomial Function and Their Multiplicity Figure 6 shows the graph of a polynomial function with four x-intercepts. Notice that at the x-intercepts, the graph must either cross the x-axis or touch the x-axis. Consequently, between consecutive x-intercepts the graph is either above the x-axis or below the x-axis.
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