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Transformations of Functions: Shifts, Reflections, and Stretching

Understand how functions are transformed through shifts, reflections, and stretching by exploring examples of vertical and horizontal shifts, reflections in the x-axis and y-axis, vertical and horizontal stretching and shrinking.

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Transformations of Functions: Shifts, Reflections, and Stretching

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  1. Example: Shift, Reflection y Example: The graph of y = x2+ 3is the graph of y = x2shifted upward three units. This is a vertical shift. 8 4 x 4 -4 -4 -8 The graphs of many functions are transformations of the graphs of very basic functions. y = x2 + 3 y = x2 The graph of y = –x2is the reflection of the graph of y = x2 in the x-axis. y = –x2

  2. Vertical Shifts y x VerticalShifts If c is a positive real number, the graph of f(x) + cis the graph of y = f(x)shifted upwardc units. If c is a positive real number, the graph of f(x) – cis the graph of y = f(x)shifted downwardc units. f(x) + c f(x) +c f(x) –c -c

  3. y 8 4 x 4 -4 -4 Example: Vertical Shifts Example:Use the graph of f(x) = |x| to graph thefunctions g(x) = |x| +3 and h(x) = |x| –4. g(x) = |x| + 3 f(x) = |x| h(x) = |x| –4

  4. Horizontal Shifts y x Horizontal Shifts If c is a positive real number, then the graph of f(x – c) is the graph of y = f(x)shifted to the rightc units. -c +c If c is a positive real number, then the graph of f(x + c) is the graph of y = f(x)shifted to the leftc units. y = f(x + c) y = f(x) y = f(x – c)

  5. y 4 4 -4 x Example: Horizontal Shifts Example:Use the graph of f(x) = x3to graphg(x) = (x – 2)3 and h(x) = (x + 4)3. f(x) = x3 g(x) = (x – 2)3 h(x) = (x + 4)3

  6. y y 4 4 x x -4 -4 (–1, –2) Example: Vertical and Horizontal Shifts Example:Graph the function using the graph of . Then a horizontal shift 5 units left. First make a vertical shift 4 units downward. (4, 2) (0, 0) (4, –2) (–5, –4) (0, – 4)

  7. Reflection in the y-Axis and x-Axis. y x The graph of a function may be a reflection of the graph of a basic function. The graph of the function y = f(–x)is the graphof y = f(x) reflected in the y-axis. y =f(–x) y = f(x) y =–f(x) The graph of the function y =–f(x) is the graph ofy = f(x)reflected in the x-axis.

  8. y y 4 4 x x 4 4 – 4 -4 Example: Reflections Example:Graph y =–(x + 3)2using the graph of y = x2. Then shiftthe graphthree units to the left. First reflectthe graphin the x-axis. y = x2 (–3, 0) y =–(x + 3)2 y =–x2

  9. y 4 is the graphof y = x2shrunk vertically by. x –4 4 Vertical Stretching and Shrinking Vertical Stretching and Shrinking If c > 1 then the graph of y= cf(x) is the graph of y = f(x) stretched vertically by c. If 0 < c < 1 then the graph of y = cf(x) is the graph of y = f(x) shrunk vertically by c. y = x2 y =2x2 Example: y =2x2 is the graph of y = x2stretched verticallyby 2.

  10. y 4 is the graph of y = |x| stretched horizontallyby . x -4 4 Horizontal Stretching and Shrinking Example: Vertical Stretch and Shrink If c > 1, the graph of y = f(cx) is the graph of y = f(x) shrunk horizontally by c. If 0 < c < 1, the graph of y = f(cx) is the graph of y = f(x) stretchedhorizontally by c. y = |2x| Example:y = |2x|is the graph of y = |x| shrunk horizontally by 2. y = |x|

  11. y y 8 8 4 4 x x -4 4 4 Step 3: -4 Step 4: Example:Graph using the graph ofy = x3. Example: Multiple Transformations Graph y = x3and do one transformation at a time. Step 1: y = x3 Step 2: y =(x + 1)3

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