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Chapter 2

Chapter 2. Radical Functions. Square Roots. In mathematics, a square root of a number a is a number y such that y 2 = a For example, 4 is a square root of 16 because 4 2 = 16 And so is -4 because (-4) 2 = 16. So what is …. = 2. … Why isn’t it 2 and -2?

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Chapter 2

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  1. Chapter 2 Radical Functions

  2. Square Roots • In mathematics, a square root of a number a is a number y such that y2 = a • For example, 4 is a square root of 16 because 42 = 16 • And so is -4 because (-4)2 = 16

  3. So what is … = 2 … Why isn’t it 2 and -2? Because means the principal square root ... … the one that isn't negative! There are two square roots, but the radical symbol means just the principal square root

  4. The square roots of 36 are … • … 6 and -6 • But = … • …6 • When you solve the equation x2 = 36, you are trying to find all possible values that might have been squared to get 36

  5. What about … = ? There is no real number that when squared is negative For the purposes of Grade 12 Pre-Calculus you cannot take the square root of a negative number

  6. Radical Function • A radical function is a function that has a variable in the radicand

  7. Order??? • Stretches and Reflections, performed in any order, followed by translations. • If is a point on then … … is a point on

  8. Graphing Radical Functions Using a Table of Values • A good place to start is to determine the domain • The radicand must be greater than or equal to zero Remember to reverse the inequality when multiplying or dividing by a negative number

  9. Graphing Radical Functions Using a Table of Values

  10. The Domain is or The Range is or

  11. Graphing Radical Functions Using Transformations

  12. Not an invariant point!! It does not map to itself!!! (0,0) maps to (1,1) (1,1) maps to (0.5, 4)

  13. Mapping Notation How points on this map to that

  14. What do you notice? has been horizontally stretched by a factor of has been horizontally stretched by a factor of and vertically stretched by a factor of 2 has been vertically stretched by a factor of 4 These all have the same graph!! They are identical!

  15. So… “a” can do anything that “b” can do and vice versa … right? Can’t we just get rid of one of them? Wrong!!! Without “a” you can’t do reflections in the x-axis and without “b” you can’t do reflections in the y-axis.

  16. Because the origin is an invariant point as far as stretching and reflecting is concerned we know that if the starting point hasn’t moved then no translations were involved. If the starting point of the graph hasn’t moved horizontally then “h” must be zero and if the starting point hasn’t shifted vertically then “k” must be zero. No amount of stretching or reflecting can change that. If the starting point has moved, then the values of h and k are just the coordinates of the place where the starting point has moved to.

  17. Remember this!!

  18. Since the graph has not been reflected in either the x or y axis we know that a and b must be positive.

  19. It can be viewed as either a purely vertical stretch or purely horizontal stretch.

  20. Viewed as a vertical stretch Viewed as a horizontal stretch

  21. Since the graph has been reflected in both the x or y axis we know that both a and b must be negative. So we can set a = -1 and find b or set b = -1 and find a.

  22. The Domain is or [-273.15, +∞) The only transformations that can change the range as compared to the base function are vertical translations and reflections over the x-axis. Neither of these occur. The Range is or [0, +∞)

  23. The graph of has been stretched vertically by a factor of about 20 and then translated horizontally about 273 units to the left.

  24. Square Root of a function First let’s look at …

  25. You can use values of to predict values of and to sketch the graph of . The domain of consists only of the values in the domain of for which The range of consists of the square roots of all the values in the range of for which is defined. Invariant points occur at a because at these values . F athe graph of i athe graph of i

  26. i The Domain of is or

  27. Invariant points occur at a because at these values .

  28. The Domain of is and the Range of is The Domain of is and the Range is

  29. Invariant points occur at (1, 1) and (1.5, 0)

  30. Find x-ints (set : First find key points of Vertex at (0, 2) and y-int = 2

  31. Set Invariant points occur at a because at these values . Invariant points occur at (-2, 0) , (2, 0) , (, 1), and (, 1)

  32. The y-coordinates of the points on are the square roots of the corresponding points on

  33. The Domain of is and the Range of is The Domain of is and the Range is

  34. The y-coordinates of the points on are the square roots of the corresponding points on

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