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Exploring Transformations of Parent Functions

In this lesson, we explore how to transform parent functions through various methods, including compressing, stretching, and reflecting shapes. We will focus on vertical and horizontal translations by adjusting constants in the function expressions. Through examples, learners will practice moving graphs up, down, left, and right, while understanding the impact of these transformations on the parent function's shape. By the end of the lesson, students will be able to sketch transformed functions and apply their knowledge to any function they encounter.

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Exploring Transformations of Parent Functions

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  1. Unit 1 Day 7 MCR 3U Feb 15, 2012 Exploring Transformations of Parent Functions

  2. a = adjusting shape (compress, stretch or reflect) c = moving up/down d = moving left/right Note: a ,c ,d  R Remember f(x) means – function with variable x Recall “Transforming”

  3. Vertical Translations • f(x) = x2 f(x) + y y 0 = x2 0 1 = x2 +1 3 = x2 + 3 2 = x2+2 x

  4. Vertical Translations • f(x) = x2 f(x) + y y 0 = x2 -1 = x2 -1 0 -3 = x2-3 -2 =x2 - 2 x Adding c to f(x) moves the graph up by c units if c is positive, down if c is negative

  5. Horizontal Translations • f(x) = x2 y y f(x + 0) = (x+0)2 f(x+1)=(x+1)2 f(x+2) =(x+2)2 f(x+3) = (x+3)2 x

  6. Horizontal Translations • f(x) = x2 y y f(x – 0) = (x-0)2 f(x-1)=(x-1)2 f(x-2) =(x-2)2 f(x-3) = (x-3)2 x • Changing a function from f(x) to f(x-d) will move the graph d units to the right. • Changing a function from f(x) to f(x+d) will move the graph d units to the left.

  7. Combining Translations • If f(x) = x2, graph f(x-2) +3: y y f(x) = x2 f(x-2)=(x-2)2 f(x-2) +3 =(x-2)2 +3 x

  8. Examples • For f(x)=x2, graph the following: • f(x) + 3 • f(x) - 1 • f(x-2) • f(x+4)

  9. Recall “Parent” functions and their “Family”

  10. Transforming Non-Quadratics • e.g. If f(x)= x , sketch f(x – 3) + 2 2 3

  11. Translating Non-Quadratics • So, for any function, if you can graph f(x), you can shift it to graph new functions! • E.g. if f(x) = 1/x, sketch f(x+2)+1 1 -2

  12. You can even be given a graph of something weird, and be told to move it! • e.g. Given f(x) below, sketch f(x+2) -1 f(x+2) -1 f(x+2) f(x)

  13. Conclusions for ALL Functions • The constants c, and d each change the location of the graph of f(x). • The shape of the graph of g(x) depends on the graph of the parent function g(x) and on the value of a. “f” represents any parent function

  14. Seatwork • Page 51#1,2,4

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