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Worksheet 1.1: Reviewing Functions 1. If f ( x) = 3 x + 7 , find a. f (12) d. f (−4) 2. If f ( x) = a. d. f ($) f (9) Math 12 b. f (a) c. f (2 x − 5) e. f (Ψ ) f. ⎛t⎞ f ⎜⎜ ⎟⎟ ⎝g⎠ b. e. f (1.8) f ( x − 4) c. f. f (−3) f (Ω) 2x , find x+4 3. Linear Functions – Consider the function f ( x) = 2 x − 8 . a. Determine the domain and range of this function. b. Determine the x and y-intercepts of this function. c. Graph this function. 4. Quadratic Functions – Consider the function f ( x) = x 2 + 3 . a. Determine the domain and range of this function. b. Does this function have a maximum and/or a minimum value? If so, what are they? c. Graph this function. x 5. Rational Functions – Consider the function f ( x) = . x +1 a. Determine the domain and range of this function b. Using your answer from part a, determine if there are any vertical and/or horizontally asymptotes for this function. c. Graph this function. 6. Radical Functions – Consider the function f ( x) = x + 3 . a. Determine the domain and range of this function. b. Are there any asymptotes? Are there any values of x and y that are not allowed? c. Graph this function. 7. Absolute-value Functions – Consider the function f ( x) = 2 x − 2 . a. Determine the domain and range of this function. b. Graph this function. 8. Inverse Functions – Consider the function f ( x) = 5 x − 13 . Write the equation for the inverse of this function. 9. Reciprocal Functions – Consider the function from question 8. Write the equation for the reciprocal of that function. Do the domain and range values change in the process? How? Worksheet 1.2: Translating Functions Math 12 1. Given that y = x , state the changes that have taken place in each of the items below. a. y = x + 3 b. y = ( x + 3) c. y = x − 12 d. y = ( x − 12) 2. Consider the function y = 3 x + 6 , we’ll call it function A. a. Convert this function to the standard form y = b( x + c) + d . Indicate any 0s and 1s. b. Move function A 3 units to the left, in standard form. c. Move function A 12 units down, in standard form. d. Move function A 8 units to the right, in standard form. e. Move function A 5 units to the left and 2 units up, in standard form. 3. Consider the function y = x 2 , we’ll call it function A. a. Convert this function to the standard form y = b( x + c) + d . Indicate any 0s and 1s. b. Translate function A 9 units to the right, in standard form. c. Move function A 7 units down, in standard form. d. Now move the function from part c 6 units to the left, then 12 units up, in standard form. 4. Consider the function y = x . State the changes that have taken place in each item. a. y = x +7 b. y = x − 12 c. y = x+ 2 −9 d. y−4= x−4 1 , we’ll call it function A. x a. Move function A 8 units down. 5. Consider the function y = b. Move function A 4 units left and 5 units up. 6. Indicate, in words, the changes that have taken place in each set of functions. a. y = x + 8 becomes y = x − 2 − 8 . b. y= c. y = 16 − ( x − 4) 2 becomes y = 16 − x 2 + 3 1 1 becomes y = + 3. x−2 x+5 Worksheet 1.3: Reflecting Functions Math 12 1. Consider the functions below. This question is strictly an exercise in drawing, not math skills. Please do not be concerned with changing the equations in any way at this point. i. Using your graphing calculator, graph each function and draw it on the grid provided. ii. No calculators - On the same grid, now graph − f (x) . iii. No calculators - Still on the same grid, now graph f (− x) . a. y= x b. y = x −1 c. y = 3− x d. 5 y = − x−3 2 e. y= 1 x +1 f. y = −( x − 2) 2 + 2 g. y= 1 3 x +1 3 h. y= i. y = x−2 j. y = ( x + 2)3 2 1 −1 x+2 2. For each function in question 1, provide the equations for − f (x) , and f (− x) . 3. Inputting the equations from question 2 into your graphing calculator, see if the graphs you drew in question 1 agree with the graphs produced by your calculator. If not, investigate the cause of the discrepancy and correct the problem. 4. For each function in question 1, a. re-graph the original function. b. Now try and visualize what the inverse or f −1 ( x) of each graph would look like. On the grid, draw the inverse. c. For each function, provide the equation for the inverse. d. Input your answers from part c into your calculator. Do the graphs agree? If not, what could be the problem? Worksheet 1.4: Stretching/Compressing Functions Math 12 1. Consider the function y = 3x 2 + 9 . We’ll call it function A. a. Horizontally compress function A by ½. b. Vertically compress function A by ½ c. Horizontally expand function A 2 times. d. Vertically expand function A 2 times. 2. If y = f (x) , then describe the changes that have taken place in each case. a. y = 3 f (2 x) b. y = −2( fx) c. y = f (2 x) d. y = f (−4 x) e. ⎛1 ⎞ y = 4 f ⎜ x⎟ ⎝2 ⎠ f. y = −2 f (4 x) 3. In Question 2, suppose that point A (-6, 12) is located on y = f (x) . As this function is stretched and/or compressed, the position of this point will also change. Determine the coordinates of this point for each step in question 2. 3. Describe the changes that have occurred to the first function to create the second function. 2x 3 a. y= x y= b. y = 6x − 3 y = −9 x − 3 c. y = x2 − 4x ⎛ x2 ⎞ y = ⎜⎜ ⎟⎟ − 2 x ⎝ 4⎠ Worksheet 1.5: Combinations of Transformations Math 12 1. Point A(4, -3) is located on y = f (x) . Indicate the coordinates of this point after each transformation: a. First, a horizontal compression of 1/2, then b. right by 4, then c. vertically expanded 5 times. 2. Repeat question 1 in this order: b, a, c. 3. Repeat question 1 in this order: a, c, b. 4. Repeat question 1 in this order: c, a, b. 5. What can you conclude from the previous questions? 6. Describe, in the proper order, the changes that have taken place to y = f (x) . a. y = 4 f ( x − 6) + 2 f. y = f (4 − x) + 5 b. y = f (−( x + 1)) − 1 g. y = −2 f ( x ) − 3 c. y = 3 f (2 x) − 6 h. y = − f ( x − 3) + 1 d. y= i. y = f (3( x + 4)) + 5 e. y = f (− x + 2) j. y = −2 f (4( x − 2)) 1 2 ⎛1 ⎞ f ⎜ x⎟ − 4 ⎝2 ⎠ 7. If point A(4, -3) is located on y = f (x) , where will this point be on y = f (2 x + 10) ? 8. If y = f (x) , what changes would have created y = f (2 x − 8) ? 9. Consider y = x 2 . This function has gone through the following transformations (but not necessarily in the order given): translation to the right by 5 units, reflection in the xaxis, vertical expansion by 4 times, translation up by 4 units, horizontal compression by ½ , and reflection in the y-axis. a. When not specified (as in this case), what is the order of transformations? b. Based on your answer from part a, write the final equation for this function after all the transformations have been performed. Note: There really is only ONE correct answer here. If your answer does not agree with that of your partner, either (or both of you) could be incorrect, so try again. Worksheet 1.6: Reciprocal/Absolute-value Transformations 1. Consider the following functions: a. b. c. d. 1 for each function. f ( x) i. Sketch the graph of y = ii. Sketch the graph of y = f (x) for each function. iii. Sketch the graph of y = f ( x ) for each function. 2. For each of the following functions, i. sketch the function. ii. sketch the graph of y = 1 . f ( x) iii. sketch the graph of y = f (x) . iv. sketch the graph of y = f ( x ) . a. f ( x) = ( x + 2) 2 − 9 b. f ( x) = −( x − 3) 2 + 4 Math 12