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Chapter 2 Radical Functions Slide 2 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 Slide 3 So what is = 2 Slide 4 Slide 5 What about Slide 6 Radical Function A radical function is a function that has a variable in the radicand Slide 7 Slide 8 Slide 9 Slide 10 Slide 11 Slide 12 Order??? Slide 13 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 Slide 14 Graphing Radical Functions Using a Table of Values Slide 15 Slide 16 Graphing Radical Functions Using Transformations Slide 17 Not an invariant point!! It does not map to itself!!! (0,0) maps to (1,1) (1,1) maps to (0.5, 4) Slide 18 Mapping Notation How points on this map to that Slide 19 What do you notice? These all have the same graph!! They are identical! Slide 20 So a can do anything that b can do and vice versa right? Cant we just get rid of one of them? Wrong!!! Without a you cant do reflections in the x-axis and without b you cant do reflections in the y-axis. Slide 21 Slide 22 Because the origin is an invariant point as far as stretching and reflecting is concerned we know that if the starting point hasnt moved then no translations were involved. If the starting point of the graph hasnt moved horizontally then h must be zero and if the starting point hasnt 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. Slide 23 Remember this!! Slide 24 Since the graph has not been reflected in either the x or y axis we know that a and b must be positive. Slide 25 It can be viewed as either a purely vertical stretch or purely horizontal stretch. Slide 26 Viewed as a vertical stretchViewed as a horizontal stretch Slide 27 Slide 28 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. Slide 29 Slide 30 Slide 31 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. Slide 32 Slide 33 Slide 34 Slide 35 Slide 36 First lets look at Square Root of a function Slide 37 Slide 38 Slide 39 Slide 40 x 00 111 42 -393 Slide 41 Slide 42 Invariant points occur at (1, 1) and (1.5, 0) Slide 43 Vertex at (0, 2) and y-int = 2 Slide 44 Slide 45 Slide 46 Slide 47 Slide 48 Slide 49 Slide 50 Slide 51 Roots, zeroes, x-intercepts, solutions ??? What is the relationship between these things? The following phrases are equivalent: "find the zeroes of f(x)" "find the roots of f(x)" "find all the x-intercepts of the graph of f(x)" "find all the solutions to f(x)=0" They are the same!!!!! Slide 52 First consider any restrictions on the variable in the radical. Slide 53 Algebraic solutions to radical equations sometimes produce extraneous roots In mathematics, an extraneous solution represents a solution that emerges from the process of solving the problem but is not a valid solution to the original problem. You must always check your solution in the original equation. Left SideRight Side Slide 54 One of the basic principles of algebra is that one can perform the same mathematical operation to both sides of an equation without changing the equation's solutions. However, strictly speaking, this is not true, in that certain operations may introduce new solutions that were not present before. The process of squaring the sides of an equation creates a "derived" equation which may not be equivalent to the original radical equation. Consequently, solving this new derived equation may create solutions that never previously existed. These "extra" roots that are not true solutions of the original radical equation are called extraneous roots and are rejected as answers. Slide 55 Slide 56 Check: Left SideRight Side Left SideRight Side The solution is x = -1 Slide 57 First consider any restrictions on the variable in the radical. Slide 58 Method 1: Slide 59 Method 2: Slide 60 Algebraic solutions sometimes produce extraneous roots, whereas graphical solutions do not produce extraneous roots. Algebraic solutions are generally exact while graphical solutions are often approximate. Slide 61 Slide 62