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TRANSFORMATIONS Shifts Stretches And Reflections Reflection about the x-axis When a function y = f(x) is multiplied by –1, the which results is f(x) reflected about the x- axis y f (x) x x x 0 0 0 1 1 -1 4 2 -2 9 3 -3 y 1( f ( x)) f ( x) Reflection about the y-axis When the argument x of a function y = f(x) is multiplied by –1, the which results is f(x) reflected about the y- axis x x x x 0 0 0 0 1 1 -1 1 4 2 -2 2 9 3 -3 3 Vertical Shift • If a positive real number k is added to a function f(x), the graph of the new function f(x)+k is shifted vertically k units f ( x) x 2 x2 x x2 1 x2 3 -2 4 7 3 -1 1 4 0 0 0 3 -1 1 1 4 0 2 4 7 3 f ( x) x 2 3 K=3 f ( x) x 2 1 K= -1 Vertical shifts f ( x) x 2 3 f ( x) f (xx2) x4 2 f ( x) x 2 f ( x) x 2 1 Right Horizontal Shift • If the argument x of a function y = f(x) is replaced by x – h for h > 0, the graph of the new function y = f(x – h) is the graph of f(x) shifted horizontally right h units. x x2 ( x 3) 2 -1 1 16 0 0 9 1 1 4 2 2 1 3 5 0 4 16 1 5 25 4 f ( x) ( x 3) 2 Left Horizontal shift • If the argument x of a function y = f(x) is replaced by x + h for h > 0, the graph of the new function y = f(x + h) is the graph of f(x) shifted horizontally left h units. f ( x) ( x 3) 2 2 x x 2 (x + 3) -5 25 4 -3 9 0 -1 1 4 0 0 9 2 4 25 4 16 49 Combining Vertical and Horizontal Shift f ( x) x 2 3 f ( x) ( x 3)2 3 f ( x) x 2 3 Vertically Stretched and Vertically Compressed Functions When a function f(x) is multiplied by a positive number a, the graph of the new function af(x) is obtained by multplying each y coordinate on the graph of y =f(x) by a. x |x| 2|x| ½|x| -1 1 2 ½ -2 2 4 1 0 0 0 0 1 1 2 ½ 2 2 4 1 Vertical Stretch When a function f(x) is multiplied by a positive number a the graph of the new function af(x) is obtained by multplying each y coordinate on the graph of y =f(x) by a. The new graph is a vertically stretched version of a f ( x) 2 x f ( x) x Vertical Compressions If the graph of a function f(x) is multiplied by a number a so that 0 < a < 1, then the new function af(x) is a vertical compression f ( x) x f ( x) 1 x 2 Shifts Vertical Name the function whose graph is given below 1. 2. 3. 4. 5. 6.