Transformations.

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Presentation transcript:

Transformations

Key Vocabulary Images Pre-images Dilations Translations

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

Vertical Shifts If c is a positive real number, the graph of f (x) + c is the graph of y = f (x) shifted upward c units. If c is a positive real number, the graph of f (x) – c is the graph of y = f(x) shifted downward c units. x y f (x) + c f (x) +c f (x) – c -c

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

Graphing Utility: Sketch the graphs given by –5 5 4 –4

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 right c units. x y -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 left c units. y = f (x + c) y = f (x) y = f (x – c)

Example: Use the graph of f (x) = x3 to graph g (x) = (x – 2)3 and h(x) = (x + 4)3 . y -4 4 f (x) = x3 h(x) = (x + 4)3 g(x) = (x – 2)3

Graphing Utility: Sketch the graphs given by –5 6 7 –1

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

The graph of a function may be a reflection of the graph of a basic function. x y The graph of the function y = f (–x) is the graph of 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 of y = f (x) reflected in the x-axis.

Example: Graph y = –(x + 3)2 using the graph of y = x2. First reflect the graph in the x-axis. Then shift the graph three units to the left. x y – 4 4 -4 x y 4 y = x2 (–3, 0) y = – (x + 3)2 y = – x2

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

Dilation: Horizontal Stretching and Shrinking 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) stretched horizontally by c. y = |2x| Example: y = |2x| is the graph of y = |x| shrunk horizontally by 2. - 4 x y 4 y = |x| is the graph of y = |x| stretched horizontally by .

Graphing Utility: Sketch the graphs given by –5 5

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