Transformations
- 1. Transformation of Functions Recognise graphs of common functions Use shifts to graph functions Use reflections to graph functions Use stretching & shrinking to graph functions Graph functions applying a sequence of transformations
- 2. The following basic graphs will be used extensively in this section. It is important to be able to sketch these from memory.
- 3. The linear function f(x) = x
- 5. The square root function
- 6. The absolute value function
- 9. We will now see how certain transformations (operations) of a function change its graph. This will give us a better idea of how to quickly sketch the graph of certain functions. The transformations are (1) translations/shifts, (2) reflections/flips, (3) stretching/change of scale.
- 10. Vertical Translations/Shifts Let f be a function and c a positive real number. The graph of y = f ( x ) + c is the graph of y = f ( x ) shifted c units vertically upward. The graph of y = f ( x ) – c is the graph of y = f ( x ) shifted c units vertically downward. y = f ( x ) + c y = f ( x ) y = f ( x ) - c y = f ( x ) c c
- 11. Vertical Translation/shift Vertical Translation For c > 0, the graph of y = f ( x ) + c is the graph of y = f ( x ) shifted up c units; the graph of y = f ( x ) c is the graph of y = f ( x ) shifted down c units.
- 12. Horizontal Translations/Shifts Let f be a function and c a positive real number. The graph of y = f (x + c) is the graph of y = f (x) shifted to the left c units. The graph of y = f (x - c) is the graph of y = f (x) shifted to the right c units. y = f ( x + c) y = f ( x ) y = f ( x - c) y = f ( x ) c c
- 13. Horizontal Translation/Shift Horizontal Translation For c > 0, the graph of y = f ( x c ) is the graph of y = f ( x ) shifted right c units; the graph of y = f ( x + c ) is the graph of y = f ( x ) shifted left c units.
- 14. Vertical shifts Moves the graph up or down Impacts only the “y” values of the function No changes are made to the “x” values Horizontal shifts Moves the graph left or right Impacts only the “x” values of the function No changes are made to the “y” values
- 15. The values that translate the graph of a function will occur as a number added or subtracted either inside or outside a function. Numbers added or subtracted inside translate left or right , while numbers added or subtracted outside translate up or down .
- 16. Recognizing the shift from the equation, examples of shifting the function f(x) = Vertical shift of 3 units up Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)
- 17. Example Use the graph of f ( x ) to obtain the graph of h ( x ) = ( x + 1) 2 – 3. Solution Step 1 Graph f ( x ) = x 2 . The graph of the standard quadratic function is shown. Step 2 Graph g ( x ) = ( x + 1) 2 . Because we add 1 to each value of x in the domain, we shift the graph of f horizontally one unit to the left. -5 -4 -3 -2 -1 1 2 3 4 5 5 4 3 2 1 -1 -2 -3 -4 -5 Step 3 Graph h ( x ) = ( x + 1) 2 – 3. Because we subtract 3, we shift the graph vertically down 3 units.
- 18. Use the basic graph to sketch the following:
- 19. Combining a vertical & horizontal shift Example of function that is shifted down 4 units and right 6 units from the original function.
- 20. Reflections The graph of f ( x ) is the reflection of the graph of f ( x ) across the x -axis. The graph of f ( x ) is the reflection of the graph of f ( x ) across the y -axis. If a point ( x , y ) is on the graph of f ( x ), then ( x , y ) is on the graph of f ( x ), and ( x , y ) is on the graph of f ( x ).
- 21. Reflecting Across x-axis (y becomes negative, -f(x)) Across y-axis (x becomes negative, f(-x))
- 22. Use the basic graph to sketch the following:
- 23. Stretching and Shrinking Graphs (Change of Scale) 10 9 8 7 6 5 4 3 2 1 1 2 3 4 -4 -3 -2 -1 h ( x ) =1/2 x 2 Let f be a function and a a positive real number. If a > 1, the graph of y = a f ( x ) is the graph of y = f ( x ) vertically stretched by multiplying each of its y -coordinates by a . If 0 < a < 1, the graph of y = a f ( x ) is the graph of y = f ( x ) vertically shrunk by multiplying each of its y -coordinates by a . f ( x ) = x 2 g ( x ) = 2 x 2
- 24. Vertical Stretching and Shrinking Change of Scale The graph of af ( x ) can be obtained from the graph of f ( x ) by stretching vertically for | a | > 1, or shrinking vertically for 0 < | a | < 1. For a < 0, the graph is also reflected across the x -axis. (The y -coordinates of the graph of y = af ( x ) can be obtained by multiplying the y -coordinates of y = f ( x ) by a .)
- 25. VERTICAL STRETCH or SHRINK Change of Scale y’s do what we think they should: If you see 3(f(x)), all y’s are MULTIPLIED by 3 (it’s now 3 times as high or low!)
- 26. Horizontal Stretching or Shrinking The graph of y = f ( cx ) can be obtained from the graph of y = f ( x ) by shrinking horizontally for |c| > 1, or stretching horizontally for 0 < | c| < 1. For c < 0, the graph is also reflected across the y -axis. (The x -coordinates of the graph of y = f ( cx ) can be obtained by dividing the x -coordinates of the graph of y = f ( x ) by c .)
- 27. Horizontal stretch & shrink We’re MULTIPLYING by an integer (not 1 or 0). x’s do the opposite of what we think they should. (If you see 3x in the equation where it used to be an x, you DIVIDE all x’s by 3, thus it’s compressed horizontally.)
- 28. Sequence of transformations Follow order of operations. 1 st transformation would be (x+2), which moves the function left 2 units (subtract 2 from each x), 2 nd transformation would be 3 times all the y’s....or change the scale by a factor of 3 3 rd transformation would be subtract 1 from all y’s,
- 29. Graph of Example
- 30. Example Use the graph of f(x) = x 3 to graph g(x) = (x+3) 3 - 4 Solution: Step 1: Because x is replaced with x+3, the graph is shifted 3 units to the left.
- 31. Example Use the graph of f(x) = x 3 to graph g(x) = (x+3) 3 - 4 Solution: Step 2: Because the equation is not multiplied by a constant, no stretching or shrinking is involved.
- 32. Example Use the graph of f(x) = x 3 to graph g(x) = (x+3) 3 - 4 Solution: Step 3: Because x remains as x, no reflecting is involved.
- 33. Example Use the graph of f(x) = x 3 to graph g(x) = (x+3) 3 - 4 Solution: Step 4: Because 4 is subtracted, shift the graph down 4 units.