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Functions: Transformations of Graphs

Functions: Transformations of Graphs. Vertical Translation: The graph of f ( x ) + k appears as the graph of f ( x ) shifted up k units ( k > 0) or down k units ( k < 0). Example 1: Sketch the graphs of. 6. 5. 4. and on the same rectangular coordinate plane. 3. 2.

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Functions: Transformations of Graphs

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  1. Functions: Transformations of Graphs Vertical Translation: The graph of f(x) + k appears as the graph of f(x) shifted upk units (k> 0) or downkunits (k< 0).

  2. Example 1: Sketch the graphs of 6 5 4 and on the same rectangular coordinate plane. 3 2 1 0 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 Functions: Transformations of Graphs The graph of f(x) is shown. Note g(x) = f(x) + 3 so the graph of g(x) is the graph of f(x) shifted up 3 units. Note h(x) = f(x) – 4 so the graph of h(x) is the graph of f(x) shifted down 4 units.

  3. Functions: Transformations of Graphs Horizontal Translation: The graph of f(x + k) appears as the graph of f(x) shifted leftk units (k> 0) or rightkunits (k< 0).

  4. Example 2: Sketch the graphs of 6 5 and on the same rectangular coordinate plane. 4 3 2 1 0 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 Functions: Transformations of Graphs The graph of f(x) is shown. Note g(x) = f(x + 4) so the graph of g(x) is the graph of f(x) shifted left 4 units. Note h(x) = f(x – 2) so the graph of h(x) is the graph of f(x) shifted right 2 units.

  5. Functions: Transformations of Graphs Reflections across the axes: The graph of - f(x) appears as the graph of f(x) reflected across the x-axis. The graph of f(- x) appears as the graph of f(x) reflected across the y-axis.

  6. Example 3: Sketch the graphs of 6 5 and 4 3 on the same rectangular coordinate plane. 2 1 0 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 Functions: Transformations of Graphs The graph of f(x) is shown. Note g(x) = - f(x) so the graph of g(x) is the graph of f(x) reflected across the x-axis. Note h(x) = f(- x) so the graph of h(x) is the graph of f(x) reflected across the y-axis.

  7. Functions: Transformations of Graphs Vertical stretches and shrinks: The graph of k f(x) appears as the graph of f(x) vertically stretched (k > 1) or vertically shrunk (0 < k < 1) by a factor of k .

  8. Example 4: Sketch the graphs of 6 and 5 4 on the same rectangular coordinate plane. 3 2 1 0 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 Functions: Transformations of Graphs The graph of f(x) is shown. Note g(x) = 2f(x) so the graph of g(x) is the graph of f(x) vertically stretched by a factor of 2. Note h(x) = 1/2 f(x) so the graph of h(x) is the graph of f(x) vertically shrunk by a factor of one-half.

  9. Functions: Transformations of Graphs Combinations of Transformations: When there are multiple transformations of a graph of a function, they should be done in this order: (1) Reflections, vertical stretches and shrinks (2) Horizontal and vertical shifts (translations)

  10. 6 5 4 3 2 1 0 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 Functions: Transformations of Graphs Example 5: Sketch the graph of f (x) = - 1/4x + 1 + 2 using transformations First, sketch the basic function y = x.

  11. 6 5 4 3 2 1 0 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 Functions: Transformations of Graphs f(x) = -1/4x + 1 + 2 Next, do the reflection across the x-axis:

  12. 6 5 4 3 2 1 0 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 Functions: Transformations of Graphs f(x) = - 1/4x + 1 + 2 Next, do the vertical shrink:

  13. Graph of f (x) = - 1/4x + 1 + 2 6 5 4 3 2 1 0 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 Functions: Transformations of Graphs f(x) = - 1/4x + 1 + 2 Last, do the horizontal and vertical shifts:

  14. 6 5 4 3 2 1 0 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 Functions: Transformations of Graphs Now try: Sketch the graph of f (x) = 2(x – 3)2 – 4 by performing transformations on the graph of another function.

  15. Functions: Transformations of Graphs END OF PRESENTATION

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