1.7 Transformations of Functions I. There are 4 basic transformations for a function f(x). y = A • f (Bx + C) + D A) f(x) + D (moves the graph + ↑ and – ↓) B) A • f(x) (this is the “slope”) 1) If | A | > 1 then it is vertically stretched. 2) If 0 < | A | < 1, then it’s a vertical shrink. 3) If A is negative, then it flips over the x-axis. C) f(x + C) (moves the graph  + and – ) D) f(Bx) or f(B(x)) (sometimes we factor out the B term) 1) If | B | > 1 then it’s a horizontal shrink. 2) If 0 < | B | < 1, then it’s horizontally stretched. 3) If B is negative, then it flips over the y-axis.

1.7 Transformations of Functions II. What each transformation does to the graph. A) f(x) f(x) + D f(x) – D B) +A f(x) +A f(x) –A f(x) . A > 1 0 < A < 1

1.7 Transformations of Functions II. What each transformation does to the graph. C) f(x) f(x + C) f(x – C) D) f(Bx) f(Bx) f(-Bx) . B > 1 0 < B < 1

1.7 Transformations of Functions III. What happens to the ordered pair (x , y) for shifts. A) f(x) + D (add the D term to the y value) Example: f(x) + 2 (5 , 4)  (5 , 6) f(x) – 3 (5 , 4)  (5 , 1) B) A • f(x) (multiply the y value by A) Example: 3 f(x) (5 , 4)  (5 , 12) ½ f(x) (5 , 4)  (5 , 2) –2 f(x) (5 , 4)  (5 , –8) C) f(x + C) (add –C to the x value) [change C’s sign] Example: f(x + 2) (5 , 4)  (3 , 4) (subtract 2) f(x – 3) (5 , 4)  (8 , 4) (add 3)

1.7 Transformations of Functions III. What happens to the ordered pair (x , y) for shifts. D) f(Bx) or f (B(x)) 1) If B > 1 (divide the x value by B) Example: f(2x) (12 , 4)  (6 , 4) f(3x) (12 , 4)  (4 , 4) f (4(x)) (12 , 4)  (3 , 4) 2) If 0<B<1 (divide the x value by B) [flip & multiply] Example: f(½x) (12 , 4)  (24 , 4) f (¾(x)) (12 , 4)  (16 , 4) 3) If B is negative (follow the above rules for dividing) Example: f(-2x) (12 , 4)  (-6 , 4) f (-½(x)) (12 , 4)  (-24 , 4)

1.7 Transformations of Functions f(x) is shown below. Find the coordinates for the following shifts. f(x) + 4 f(x) – 6 (-4,6) (-1,4) (1,7 ) (2,1) (-4,-4) (-1,-6) (1,-3) (2,-9) f(x + 4) f(x – 3) (-8,2) (-5,0) (-3,3) (-2,-3) (-1,2) (2,0) (4,3) (5,-3) 2 f(x) ½ f(x) -3 f(x) (-4,4) (-1,0) (1,6) (2,-6) (-4,1) (-1,0) (1,3/2) (2,-3/2) (-4,-6) (-1,0) (1,-9) (2,9) f(2x) f(½x) f(-3(x)) (-2,2) (-1/2,0) (1/2,3) (1,-3) (-8,2) (-2,0) (2,3) (4,-3) (4/3,2) (1/3,0) (-1/3,3) (-2/3,-3) Homework: page 79 # 1, 2, 5 – 17 odd