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Section 3.3

Section 3.3. Graphing Techniques: Transformations. Horizontal and Vertical Shifts. We investigated what the graph Let’s also graph and. Horizontal and Vertical Shifts (cont.).

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Section 3.3

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  1. Section 3.3 Graphing Techniques: Transformations

  2. Horizontal and Vertical Shifts • We investigated what the graph • Let’s also graph and

  3. Horizontal and Vertical Shifts (cont.) • We can notice that the graph of g(x) looks very similar to f(x) but moved on the coordinate plane. • How did it shift? • Up two spaces • h(x) also looks similar to f(x) but shifted. • How did it shift? • Right one space • f(x) is called the “Parent Function”

  4. Vertical Shift • Given the graph of some parent function f(x)… • To graph f(x) + c • Shift c units upward • To graph f(x) – c • Shift c units downward Adding or subtracting a constant outside the parent function corresponds to a vertical shift that goes with the sign. Examples of “outside the function”

  5. Horizontal Shift • Given the graph of some parent function f(x)… • To graph f(x + c) • Shift c units left • To graph f(x – c) • Shift c units right Adding or subtracting a constant inside the parent function corresponds to a horizontal shift that goes opposite the sign. Examples of “inside the function”

  6. Explain how this graph shifts compared to its parent graph Up 2 Right 3 Down 4 Left 2

  7. Given the parent function f(x) = x2, write a new equation with the following shifts. • Shift up 4 units • Add 4 units outside the function • f(x) + 4 = x2 + 4 • Shift right 1 unit • Subtract 1 unit inside the function • f(x – 1) = (x – 1)2 • Shift down 3 units, and left 2 units • Subtract 3 units outside, and add 2 units inside the function • f(x+2) – 3 = (x + 2)2 – 3

  8. Reflections about the Axes • Let’s look at the graph of again. • Now graph

  9. Reflection • Note the graph of is reflected about the x-axis, and the result is the graph of • So with a given function f(x), to flip over the x – axis, use –f(x) • A negative symbol in front of the parent graph flips the graph over the x-axis. • This should make sense from Chapter 2. When reflecting over the x-axis, we flip the sign of all y values, which is exactly what we did in the previous example.

  10. Reflection • Similarly, when reflecting over the y – axis, we simply replace x with –x. • i.e. g(x) = f(-x) • Graph and • So to flip f(x) over the y-axis, evaluate f(-x) • A negative ON the x flips the graph over the y-axis.

  11. Examples of Reflection • Reflect over x-axis • Reflect over y-axis

  12. When shifting a graph… • Follow this order: • 1. Horizontal Shifts • 2. Reflection over x or y axis • 3. Vertical Shifts

  13. Stretching and Compressing • Let’s look at the graph of • Now graph and

  14. Vertical Stretch and Compress • The graph of is found by • Vertically stretching the graph of f(x) • If c > 1 • Vertically compressing the graph of f(x) • If 0 < c < 1

  15. Write the function whose graph is the graph of f(x) = x3 with the following transformations. • Vertically Stretched by a factor of 2 • J(x) = 2x3 • Reflected about the y – axis • G(x) = (-x)3 • Vertically compressed by a factor of 3 • H(x) = (1/3)x3 • Shifted left 2 units, reflected about x – axis • K(x) = -(x + 2)3

  16. Use the given graph to sketch the indicated functions. • y = f(x + 2)

  17. Use the given graph to sketch the indicated functions. • y = -f(x – 2)

  18. Use the given graph to sketch the indicated functions. • y = 2f(–x)

  19. Sketch the graphs of the following functions using horizontal and vertical shifting. • g(x) = x2 + 2 • The 2 is being added “outside” the function • Shifts up 2 units from parent function f(x) = x2

  20. Sketch the graphs of the following functions using horizontal and vertical shifting. • h(x) = (x + 2)2 • The 2 is being added “inside” the function • Shifts 2 units left from parent function f(x) = x2

  21. Sketch the graphs of the following functions using horizontal and vertical shifting. • g(x) = (x – 3)2 + 2 • Shifts right 3 units and up 2 units from f(x) = x2

  22. Sketch G(x) = -(x + 2)2 • Start with parent graph f(x) = x2 • Shift the graph 2 units left to obtain f(x + 2) = (x + 2)2 • Reflect over the x – axis to obtain –f(x+2) = -(x + 2)2

  23. Graph

  24. Horizontal Stretch and Compress • Similarly when multiplying by a constant c “inside” the function • The graph of f(cx) is found by: • Horizontally stretching the graph of f(x) • If 0 < c < 1 • Horizontally compressing the graph of f(x) • c > 1

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