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Pre-Calculus

Pre-Calculus . Chapter 1 Functions and Their Graphs. x. x. Warm Up 1.4. A Norman window has the shape of a square with a semicircle mounted on it. Find the width of the window if the total area of the square and the semicircle is to be 200 ft 2 . 1.4 Transformation of Functions.

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Pre-Calculus

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  1. Pre-Calculus Chapter 1 Functions and Their Graphs

  2. x x Warm Up 1.4 • A Norman window has the shape of a square with a semicircle mounted on it. Find the width of the window if the total area of the square and the semicircle is to be 200 ft2.

  3. 1.4 Transformation of Functions • Objectives: • Recognize graphs of common functions. • Use vertical and horizontal shifts and reflections to graph functions. • Use nonrigid transformations to graph functions.

  4. Vocabulary • Constant Function • Identity Function • Absolute Value Function • Square Root Function • Quadratic Function • Cubic Function • Transformations of Graphs • Vertical and Horizontal Shifts • Reflection • Vertical and Horizontal Stretches & Shrinks

  5. Common Functions • Sketch graphs of the following functions: • Constant Function • Identity Function • Absolute Value Function • Square Root Function • Quadratic Function • Cubic Function

  6. Constant Function

  7. Identity Function

  8. Absolute Value Function

  9. Square Root Function

  10. Quadratic Function

  11. Cubic Function

  12. Exploration 1 • Graph the following functions in the same viewing window: y = x2 + c, where c =–2, 0, 2, and 4. Describe the effect that c has on the graph.

  13. Exploration 2 • Graph the following functions in the same viewing window: y = (x + c)2, where c= –2, 0, 2, and 4. Describe the effect that c has on the graph.

  14. Vertical and Horizontal Shifts • Let c be a positive real number. Shifts in the graph of y = f (x)are as follows: • h(x) = f (x) + c______________________ • h(x) = f (x) – c______________________ • h(x) = f (x – c)______________________ • h(x) = f (x + c)______________________

  15. Example 1 Compare the graph of each function with the graph of f (x) = x3without using your graphing calculator. • g(x) = x3 – 1 • h(x) = (x – 1)3 • k(x) = (x + 2)3 + 1

  16. Example 2 • Use the graph of f (x) = x2 to find an equation for g(x) and h(x).

  17. Exploration 3 • Compare the graph of each function with the graph of f (x) = x2by using your graphing calculator to graph the function and f in the same viewing window. Describe the transformation. • g(x) = –x2 • h(x) = (–x)2

  18. Reflections in the Coordinate Axes • Reflections in the coordinate axes of the graph of y = f (x)are represented as follows: • h(x) = –f (x) _______________________ • h(x) = f (–x)_______________________

  19. Example 3 • Use the graph of f (x) = x4 to find an equation for g(x) and h(x).

  20. Example 4 Compare the graph of each function with the graph of

  21. Exploration 4 • Graph the following functions in the same viewing window: y = cx3, where c = 1, 4and ¼. Describe the effect that c has on the graph.

  22. Exploration 5 • Graphing the following functions in the same viewing window: y = (cx)3, where c = 1, 4and ¼. Describe the effect that c has on the graph.

  23. Nonrigid Transformations • Rigid Transformation Changes position of the graph but maintains the shape of the original function. • Horizontal or vertical shifts and reflections. • Nonrigid Transformation Causes a distortion in the graph. Changes the shape of the original graph. • Vertical or horizontal stretches and shrinks.

  24. Vertical Stretch or Shrink • Original function y = f (x). • Transformation y = cf (x). Each y-value is multiplied by c. • Vertical stretch if c > 1. • Vertical shrink if 0 < c < 1.

  25. Horizontal Stretch or Shrink • Original function y = f (x). • Transformation y = f (cx). Each x-value is multiplied by 1/c. • Horizontal shrink if c > 1. • Horizontal stretch if 0 < c < 1.

  26. Homework 1.4 • Worksheet 1.4 • #5, 7, 11, 13, 16, 20, 24, 26, 27, 33, 37, 39, 42, 45, 47, 51, 53, 57, 61, 63, 67

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