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Understanding Symmetry and Transformations in Functions

This section delves into the concepts of symmetry in functions, illustrating how graphs behave in relation to the x-axis, y-axis, and the origin. It explains even and odd functions, including their definitions and algebraic tests for symmetry. Additionally, transformation rules for basic functions are detailed, including vertical and horizontal shifts, reflections, and vertical stretching or shrinking. This comprehensive overview is essential for understanding how to manipulate and analyze functions graphically.

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Understanding Symmetry and Transformations in Functions

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  1. Chapter 1 Section 1.7 Symmetry & Transformations

  2. Points and Symmetry

  3. Types of Symmetry Symmetry with respect to the • x-axis (x, y) & (x, -y) are reflections across the x-axis • y-axis (x, y) & (-x, y) are reflections across the y-axis • Origin (x, y) & (-x, -y) are reflections across the origin

  4. Even and Odd Functions • Even Function: graph is symmetric to the y-axis • Odd Function: graph is symmetric to the origin • Note: Except for the function f(x) = 0, a function can not be both even and odd.

  5. Algebraic Tests of Symmetry/Tests for Even & Odd Functions • f(x) = - f(x) symmetric to x-axis neither even nor odd (replace y with –y) • f(x) = f(-x) symmetric to y-axis even function (replace x with –x) • - f(x) = f(-x) symmetric to origin odd function (replace x with –x and y with –y)

  6. Basic Functions

  7. Basic Functions

  8. Basic Functions

  9. Basic Functions

  10. Basic Functions

  11. Basic Functions

  12. Basic Functions

  13. Transformations withthe Squaring Function

  14. Transformations with theAbsolute Value Function

  15. Transformation Rules • EquationHow to obtain the graph For (c > 0) • y = f(x) + c Shift graph y = f(x) up c units • y = f(x) - c Shift graph y = f(x) down c units • y = f(x – c) Shift graph y = f(x) right c units • y = f(x + c) Shift graph y = f(x) left c units

  16. Transformation Rules • EquationHow to obtain the graph • y = -f(x) (c > 0) Reflect graph y = f(x) over x-axis • y = f(-x) (c > 0) Reflect graph y = f(x) over y-axis • y = af(x) (a > 1) Stretch graph y = f(x) vertically by factor of a • y = af(x) (0 < a < 1) Shrink graph y = f(x) vertically by factor of a Multiply y-coordinates of y = f(x) by a

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