290 CHAPTER 2 Graphs and Functions Summary of GraphingTechniques Assume a 70, h 70, and k 70. In comparison with the graph of y = ƒ1x2: 1. The graph of y =ƒ1x2 +k is translated up k units. 2. The graph of y =ƒ1x2 −k is translated down k units. 3. The graph of y =ƒ1x +h2 is translated to the left h units. 4. The graph of y =ƒ1x −h2 is translated to the right h units. 5. The graph of y =a ƒ1x2 is a vertical stretching of the graph of y = ƒ1x2 if a 71. It is a vertical shrinking if 0 6a 61. 6. The graph of y =ƒ1ax2 is a horizontal stretching of the graph of y = ƒ1x2 if 0 6a 61. It is a horizontal shrinking if a 71. 7. The graph of y = −ƒ1x2 is reflected across the x-axis. 8. The graph of y =ƒ1 −x2 is reflected across the y-axis. CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. 1. To graph the function ƒ1x2 = x2 - 3, shift the graph of y = x2 down units. 2. To graph the function ƒ1x2 = x2 + 5, shift the graph of y = x2 up units. 3. The graph of ƒ1x2 = 1x + 422 is obtained by shifting the graph of y = x2 to the 4 units. 4. The graph of ƒ1x2 = 1x - 722 is obtained by shifting the graph of y = x2 to the 7 units. 5. The graph of ƒ1x2 = -2x is a reflection of the graph of y = 2x across the -axis. 6. The graph of ƒ1x2 = 2-x is a reflection of the graph of y = 2x across the -axis. 7. To obtain the graph of ƒ1x2 = 1x + 223 - 3, shift the graph of y = x3 to the left units and down units. 8. To obtain the graph of ƒ1x2 = 1x - 323 + 6, shift the graph of y = x3 to the right units and up units. 9. The graph of ƒ1x2 = 0 -x 0 is the same as the graph of y = 0 x 0 because reflecting it across the -axis yields the same ordered pairs. 10. The graph of x = y2 is the same as the graph of x = 1-y22 because reflecting it across the -axis yields the same ordered pairs. 11. Concept Check Match each equation in Column I with a description of its graph from Column II as it relates to the graph of y = x2. 2.7 Exercises I (a) y = 1x - 722 (b) y = x2 - 7 (c) y = 7x2 (d) y = 1x + 722 (e) y = x2 + 7 II A. a translation to the left 7 units B. a translation to the right 7 units C. a translation up 7 units D. a translation down 7 units E. a vertical stretching by a factor of 7
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