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Chapter 2 Radical Functions

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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

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So what is … = 2

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What about …

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

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Order???

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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

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Graphing Radical Functions Using a Table of Values

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Graphing Radical Functions Using Transformations

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

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Mapping Notation How points on this map to that

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What do you notice? These all have the same graph!! They are identical!

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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.

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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.

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Remember this!!

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

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

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Viewed as a vertical stretchViewed as a horizontal stretch

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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.

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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.

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

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x 00 111 42 -393

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

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Vertex at (0, 2) and y-int = 2

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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!!!!!

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First consider any restrictions on the variable in the radical.

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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

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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.

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Check: Left SideRight Side Left SideRight Side The solution is x = -1

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First consider any restrictions on the variable in the radical.

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Method 1:

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Method 2:

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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.

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