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ch.6 Transformations of Functions and Their Graphs

ch.6 Transformations of Functions and Their Graphs -...

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Chapter 6 Transformations of Functions and Their Graphs 6.1 Vertical and Horizontal Shifts

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Key Points Horizontal and vertical graphical shifts Finding a formula for a shifted graph in terms of the formula for the original graph
The graphs of many functions are transformations of the graphs of very basic functions. The graph of y = – x 2 is the reflection of the graph of y = x 2 in the x -axis. Example : The graph of y = x 2 + 3 is the graph of y = x 2 shifted upward three units . This is a vertical shift . x y -4 4 4 -4 -8 8 y = x 2 y = x 2 + 3 y = x 2

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f ( x ) f ( x ) + c + c f ( x ) c -c If c is a positive real number, the graph of f ( x ) + c is the graph of y = f ( x ) shifted upward c units. Vertical Shifts 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
h ( x ) = |x| 4 Example : Use the graph of f ( x ) = |x| to graph the functions g ( x ) = |x| + 3 and h ( x ) = |x| 4. f ( x ) = |x| x y -4 4 4 -4 8 g ( x ) = |x| + 3

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

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-4 y 4 x -4 x y 4 Example : Graph the function using the graph of . First make a vertical shift 4 units downward . Then a horizontal shift 5 units left . 4 5 - + = x y x y = (0, 0) (4, 2) (0, – 4) (4, –2) x y = 4 - = x y 4 5 - + = x y (– 5, –4) (–1, –2)
Vertical Shift If g ( x ) is a function and k is a positive constant, then the graph of y = g ( x ) + k is the graph of y = g (x) shifted vertically upward by k units. y = g ( x ) − k is the graph of y = g ( x ) shifted vertically downward by k units. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

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Vertical Shift: The Heating Schedule for an Office Building Example 1 To save money, an office building is kept warm only during business hours. At midnight ( t = 0), the building’s temperature ( H ) is 50 F . This temperature is maintained until 4 am. Then the building begins to warm up so that by 8 am the temperature is 70 F. At 4 pm the building begins to cool. By 8 pm, the temperature is again 50 F. Suppose that the building’s superintendent decides to keep the building 5 F warmer than before. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally 4 8 1 2 1 6 2 0 2 4 5 0 5 5 6 0 6 5 7 0 7 5 4 8 1 2 1 6 2 0 2 4 5 0 5 5 6 0 6 5 7 0 7 5 Graph of the original heating schedule Graph of the new heating schedule obtained by shifting original graph upward by 5 units. t (hours after midnight) t H ( F) H ( F) H = f ( t ) H = p ( t ), new schedule H = f ( t ), original schedul e
Vertical Shift: The Heating Schedule for an Office Building Example 2 What is the relationship between the formula for f ( t ), the original heating schedule and p ( t ), the new heating schedule?

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ch.6 Transformations of Functions and Their Graphs -...

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