The graph approaches zero as x approaches positive infinity so = 0. 1-5 Parent Functions and Transformations Describe the following characteristics of the graph of each parent function: domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing. The graph is decreasing on (− , 0) and (0, ). 5. f (x) = c SOLUTION: 2. f (x) = SOLUTION: The graph is continuous for all values of x, so D = {x |x }, The only y value in the domain is c. Thus, R = {y | y = c, c }. If c = 0, all real numbers are x-intercepts. If c ≠ 0, there are no x-intercepts. The graph is continuous for all values of x except when x = 0, so D = {x | x ≠ 0, x . The range includes all values of y except when x = }. 0, so R = {y | y ≠ 0, y The graph does not cross the x- or y-axis and thus has no intercepts. The graph intersects the y-axis at (0, c), so there is a y-intercept at (0, c). If c 0, the graph is symmetric with respect to the y-axis. If c = 0, the graph is symmetric with respect to the x-axis, y-axis, and origin. The graph is continuous. The graph has mirror images over the origin, thus he graph is symmetric with respect to the origin. The graph approaches c as x approaches negative infinity so . The graph has an infinite discontinuity at x = 0. The graph approaches zero as x approaches negative infinity so = 0. The graph approaches c as x approaches infinity so . The graph approaches zero as x approaches positive infinity so = 0. The graph is constant on (− The graph is decreasing on (− 5. f (x) = c SOLUTION: eSolutions Manual - Powered by Cognero , 0) and (0, ). Use the graph of f (x) = function. 8. g(x) = , ). to graph each SOLUTION: g(x) = f (x + 3). Therefore, g(x) is the graph of f (x) = 3 units to the left. translated Page 1 The graph approaches c as x approaches infinity so . 1-5 Parent Functions and Transformations The graph is constant on (− , Use the graph of f (x) = function. 8. g(x) = ). to graph each 14. g(x) = SOLUTION: SOLUTION: g(x) = f (x + 7) − 4. g(x) = f (x + 3). Therefore, g(x) is the graph of f (x) = 3 units to the left. Use the graph of f (x) = –4 Therefore, g(x) is the graph of f (x) = translated translated 7 units left and 4 units down. Describe how the graphs of f (x) = [[x]] and g(x) are related. Then write an equation for g(x). to graph each function. 11. g(x) = +4 SOLUTION: g(x) = f (x) + 4. Therefore, g(x) is the graph of f (x) = translated 4 17. SOLUTION: There are several important characteristics for f (x) = [[x]]. First determine if the graph increasing from left to right. Identify if the graph has open dots on the left or right. Determine the length of each horizontal line. Also identify how far a horizontal segment on the x-axis or y -axis is from the origin. units up. The graph of g(x) is decreasing from left to right which is the opposite or a reflection of f (x) = [[x]]. 14. g(x) = The graph of g(x) has closed dots on the left and open on the right which is the opposite as f (x) = [[x]]. –4 SOLUTION: g(x) = f (x + 7) − 4. Therefore, g(x) is the graph of f (x) = translated 7 Each horizontal line is 1 unit which is the same as f (x) = [[x]]. units left and 4 units down. eSolutions Manual - Powered by Cognero The horizontal bar on the x-axis is shifted five units to the right. Thus the graph of g(x) is the graph of f (x) reflected in the y -axis and translated 5 units right Page 2 when g(x) = [[5 – x]], or reflected in the y -axis and translated 5 units up when g(x) = [[−x]] + 5. (x) reflected in the y -axis and translated 5 units right when g(x) = [[5 – x]], or reflected in the y -axis and translated 5 units up when g(x) = [[−x]] + 5. 1-5 Parent Functions and Transformations Describe how the graphs of f (x) = |x| and g(x) are related. Then write an equation for g(x). Describe how the graphs of f (x) = [[x]] and g(x) are related. Then write an equation for g(x). 20. 17. SOLUTION: SOLUTION: There are several important characteristics for f (x) = [[x]]. First determine if the graph increasing from left to right. Identify if the graph has open dots on the left or right. Determine the length of each horizontal line. Also identify how far a horizontal segment on the x-axis or y -axis is from the origin. The central characteristic of f (x) = x is the point where the two lines meet. For our purposes here, it can be considered as a vertex or a critical point. This point is at (0, 0) for the parent function. Identifying where it shifts will help you identify g(x). Note that these are translations only. For reflections and dilations, we will have to consider more aspects of the graph. The graph of g(x) is decreasing from left to right which is the opposite or a reflection of f (x) = [[x]]. The vertex is at (8, 0), so the vertex is translated 8 unit to the right. Therefore, the graph of f (x) is also translated 8 unit to the right. The graph of g(x) has closed dots on the left and open on the right which is the opposite as f (x) = [[x]]. Now we need to identify an equation for g(x). The x-coordinate tells us what changed inside the absolute value symbols. Treat this like a zero for a linear equation. If the coordinate is 8, the expression inside the absolute value should be x − 8. Each horizontal line is 1 unit which is the same as f (x) = [[x]]. The horizontal bar on the x-axis is shifted five units to the right. Thus the graph of g(x) is the graph of f (x) reflected in the y -axis and translated 5 units right when g(x) = [[5 – x]], or reflected in the y -axis and translated 5 units up when g(x) = [[−x]] + 5. Thus, the graph of g(x) is the graph of f (x) translated 8 units to the right; g(x) = | x – 8 |. Describe how the graphs of f (x) = |x| and g(x) are related. Then write an equation for g(x). 23. SOLUTION: 20. SOLUTION: The central characteristic of f (x) = x is the point our purposes here, it can be considered as a vertex or a critical point. This point is at (0, 0) for the parent function. Identifying eSolutions Manual Powered Cognero where the -two linesbymeet. For The central characteristic of f (x) = x is the point where the two lines meet. For our purposes here, it can be considered as a vertex or a critical point. This point is at (0, 0) for the parent function. Identifying where it shifts will help you identify g(x). Note that these are translations only. For reflections and dilations, we will have to consider more aspects of Page 3 the graph. The vertex is at (1, −2), so the vertex is translated 1 linear equation. If the coordinate is 8, the expression inside the absolute value should be x − 8. the absolute value symbols. It describes the vertical shift from the origin. The y-coordinate is −2, so we need to subtract 2. Thus, the graph of g(x) is the graph of f (x) translated 1-5 8Parent Functions units to the right; g(x) =and | x – Transformations 8 |. g(x) = | x − 1 | − 2 Identify the parent function f (x) of g(x), and describe how the graphs of g(x) and f (x) are related. Then graph f (x) and g(x) on the same axes. 26. g(x) = 23. SOLUTION: g(x) = 4f (x + 1), so the graph of g(x) is the graph of f(x) translated 1 unit to the left and expanded vertically. The translation left is represented by the addition of 1 on the inside of f (x). The expansion is represented by the coefficient of 4 on the outside of f(x). SOLUTION: The central characteristic of f (x) = x is the point where the two lines meet. For our purposes here, it can be considered as a vertex or a critical point. This point is at (0, 0) for the parent function. Identifying where it shifts will help you identify g(x). Note that these are translations only. For reflections and dilations, we will have to consider more aspects of the graph. The vertex is at (1, −2), so the vertex is translated 1 unit to the right and 2 units down. Therefore, the graph of f (x) is also translated 1 unit to the right and 2 units down. Now we need to identify an equation for g(x). 29. g(x) = −2| x + 5| The x-coordinate tells us what changed inside the absolute value symbols. Treat this like a zero for a linear equation. If the coordinate is 1, the expression inside the absolute value should be x − 1. SOLUTION: g(x) = −2f (x + 5), so g(x) is the graph of f (x) translated 5 units to the left, expanded vertically, and reflected in the x-axis. The translation left is represented by the addition of 5 on the inside of f (x). The expansion is represented by the coefficient of 2 on the outside of f (x). The reflection is represented by the negative coefficient on the outside of f (x). The y-coordinate tells us what was added outside of the absolute value symbols. It describes the vertical shift from the origin. The y-coordinate is −2, so we need to subtract 2. g(x) = | x − 1 | − 2 Identify the parent function f (x) of g(x), and describe how the graphs of g(x) and f (x) are related. Then graph f (x) and g(x) on the same axes. 26. g(x) = Graph each function. SOLUTION: g(x) = 4f (x + 1), so the graph of g(x) is the graph of f(x) translated 1 unit to the left and expanded vertically. The translation left is represented by the addition of 1 on the inside of f (x). The expansion is represented by the coefficient of 4 on the outside of f(x). eSolutions Manual - Powered by Cognero 32. SOLUTION: Page 4 f(−2) ≠ −4 and f (7) ≠ 3, so the points (−2, −4) and (7, 3) are not included in the graph. Draw circles at these points. 1-5 Parent Functions and Transformations Graph each function. 35. 32. SOLUTION: SOLUTION: 2 On the interval [−∞, −2), graph y = −x . On the interval [−2, 7), graph y = 3. 2 On the interval [7, ∞), graph y = (x − 5) . Multiple points must be found for x = −2 and x = 7 because of the domain intervals. Since f (−2) = 3 and f (7) = 6, draw dots at (−2, 3) and (7, 6). f(−2) ≠ −4 and f (7) ≠ 3, so the points (−2, −4) and (7, 3) are not included in the graph. Draw circles at these points. On the interval (−∞, −3), graph y = |x − 5|. On the interval [−1, 3), graph y = 4x − 3. On the interval [4, ∞), graph y = . Since f (−1) = −7 and f (4) = 2, draw dots at (−1, −7) and (4, 2). f(−3) ≠ −8 and f (3) ≠ 9, so the points (−3, −8) and (3, 9) are not included in the graph. Draw circles at these points. 38. POSTAGE The cost of a first-class postage stamp in the U.S. from 1988 to 2008 is shown in the table below. Use the data to graph a step function. 35. SOLUTION: SOLUTION: Let x = 0 represent 1988. The price changes at the beginning of the year, so the dots will appear on the left end of each segment. Plot each of the following segments. On the interval (−∞, −3), graph y = |x − 5|. On the interval [−1, 3), graph y = 4x − 3. On the interval [4, ∞), graph y eSolutions Manual - Powered by Cognero = . Since f (−1) = −7 and f (4) = 2, draw dots at (−1, −7) x-coordinates dot circle 0 3 3 7 7 11 y-coordinates 0.25 0.29 0.32 Page 5 and (4, 2). 1-5 f(−3) ≠ −8 and f (3) ≠ 9, so the points (−3, −8) and (3, 9) are not Functions included in theand graph. Draw circles at Parent Transformations these points. 38. POSTAGE The cost of a first-class postage stamp in the U.S. from 1988 to 2008 is shown in the table below. Use the data to graph a step function. Use the graph of f (x) to graph g(x) = |f (x)| and h (x) = f (|x|). 41. f (x) = SOLUTION: |f (x)| replaces all of the negative y-values with the corresponding positive y-values. If f (–3) = , then |f (–3)| = . To graph g(x) = |f (x)|, reflect the range with respect to the x-axis for all elements of the domain where f (x) is less than zero. f(|x|) replaces all of the negative x-values with the SOLUTION: Let x = 0 represent 1988. The price changes at the beginning of the year, so the dots will appear on the left end of each segment. Plot each of the following segments. x-coordinates dot circle 0 3 3 7 7 11 11 13 13 14 14 18 18 19 19 20 20 y-coordinates corresponding positive x-values. If f (–3) = , then f(|–3|) = . To graph h(x) = f (|x|), replace the range for x < 0 with a reflection of the range for x > 0 with respect to the y-axis. 0.25 0.29 0.32 0.33 0.34 0.37 0.39 0.41 0.42 44. f (x) = 3 2 x + 2x – 8x – 2 SOLUTION: Use the graph of f (x) to graph g(x) = |f (x)| and h (x) = f (|x|). eSolutions Manual - Powered by Cognero 41. f (x) = SOLUTION: Page 6 |f (x)| replaces all of the negative y-values with the corresponding positive y-values. If f (–7) = 1-5 Parent Functions and Transformations |f (x)| replaces all of the negative y-values with the corresponding positive y-values. If f (–7) = , then |f (–3)| = . To graph g(x) = |f (x)|, reflect the range with respect to the x-axis for all elements of the domain where f (x) is less than zero. a. Write a greatest integer function f (x) that would represent the cost for x units of cab fare, where x > 0. Round to the nearest unit. b. Graph the function. c. How would the graph of f (x) change if the fare for the first unit increased to $3.70 while the cost per unit remained at $0.40? Graph the new function. SOLUTION: a. When there is only a fraction of a unit, we must round up. For example, if 3.4 units are used, the customer will be charged for 4 units. To accomplish this, use [[x + 1]] when x is not a whole number. b. f (|x|) replaces all of the negative x-values with the corresponding positive x-values. If f (–3) = = , then f (|–3|) . To graph h(x) = f (|x|), replace the range for x < 0 with a reflection of the range for x > 0 with respect to the y-axis. c. If the fare for one unit increased to $3.70, and the cost per unit was still $0.40, then the cost per trip must have increased to $3.30. This will cause a vertical translation of the graph of $0.80. The graph of f (x) is translated 0.8 unit up. 47. TRANSPORTATION In New York City, the standard cost for taxi fare is shown. One unit is equal to a distance of 0.2 mile or a time of 60 seconds, when the car is not in motion. a. Write a greatest integer function f (x) that would represent the cost for x units of cab fare, where x > 0. Round to the nearest unit. b. Graph the function. eSolutions Manual - Powered by Cognero Page 7