Loudoun County Public Schools / Overview



Name:______________Block:______________Algebra2 Unit 6Radical Functions1476375114935Unit 6: Radical FunctionsDay 1: Simplifying Nth roots Day 2: Solving Power & Radical Equations Day 3: Review for quizDay 4: QuizDay 5: Inverse Functions; Graphing Square & Cube Root FunctionsDay 6: Graphing Square & Cube Root FunctionsDay 7: Review for TestDay 8: Unit TEST(Check syllabus & website for quiz and test dates)Day 1: Simplifying Nth rootsIn these notes we will learn how to simplify nth rootsSo we can solve power equations.What does it mean to be a "perfect square"?Write the following as squares (power of 2). Then take the square root. How are roots and radicals related?Number or variable Written as a squareInterpreted as a root366?6=(6)21681 Think about the denominator…What does it mean to be a "perfect cube"?Write the following as cubes (power of 3). Then take the cube root. How are roots and radicals related?Number or variable Written as a squareInterpreted as a root82?2?2=(2)32764 Think about the denominator…Rational exponents are another way of writing radicals – think POWER over ROOTExponential NotationRadical Notation()2 or ()3 or POWER (numerator) Base (x)ROOT (denominator)Index (Root) Exponent (Power)Radicand (Base)CONVERTING (REWRITING) Rational Exponents to Radical Notation and back!How would you write in radical notation?How would you write in radical notation?How would you write in exponential notation? How would you write in exponential notation? EVALUATING Rational Exponents and Radical Notation = = = = Evaluate the expression without using the calculator. Use your Exponent Chart ** You will eventually take a NO calculator section on tests…so start memorizing and recognizing your perfect squares and cubes!!!!Writing Radicals in Simplest Form – use your Exponent Chart for perfect nth Roots Find perfect 3rd Root Find perfect 4th RootOperations with Radicals: Addition and Subtraction - Need Multiplication - NeedDivision - NeedWhen we are left with radicals in the denominator… we ____________________________!With square roots: NOW with other roots …The goal is to get a perfect nth root in the denominator to get rid of the radical We need a perfect cube in the denominator… 4 goes into which perfect cube? _______ So I need to multiply by… _______ We need a perfect cube in the denominator… 3 goes into which perfect cube? _______ So I need to multiply by… _______ We need a perfect cube in the denominator… 2 goes into which perfect cube? _______ So I need to multiply by… _______ We need a 4th root in the denominator… 8 goes into which x4 from our table? _______ So I need to multiply by… _______Simplify Expressions with Variables – Factor nth Roots. Leave answers in radical notation.Day 2: Solving Power & Radical EquationsIn these notes we will learn how to solve Power & Radical Equations. What skills do you think this will require?Solving power equations: A power equation is an equation that involves a variable raised to a power. Steps to solving a Power EquationIsolate the exponent first.Take the appropriate root of both sides. Even Roots: Odd Roots: use the signUse i when you are taking the SQUARE root of a NEGATIVE #Check all solutions by plugging your answer into the original equation.Write an example of a power equation: When solving, we must follow the reverse order of operations.1. Add or subtract 2. Multiply or divide 3. Exponents 4. Parentheses Recall from quadratics…the Square root method: (x + 3) 2 = 64 3 (x – 5)2 = - 27Note: Whenever we solve by square rooting a number, we must put a ______ in front!This is true for ALL even roots (ie: 4th root, 6th root), but NOT true for odd roots. Why is this? 1. Give me a # I can square to get 4. 2. Give me a complex # I can square to get -4.3. Give me a # I can cube to get 8.4. Give me a # I can cube to get -8.5. Give me a # I can raise to the 4th power to get 81. 2x4 = 162 (x – 2) 3 = -125 6x3 = 384 (x – 3) 4 = 625 x4 = 32 2x3 = x3 + 54 (x + 1) 5 = 100 (x – 1)5 – 3 = –35 2(x – 9)3 = 250Solving radical equations: A radical equation is an equation that involves a variable in the radicand. (Part under the radical symbol). Steps to solving radical equations: Isolate the radical on one side of the equation!Raise each side of the equation to the same power to eliminate the radical. You will be left with a linear, quadratic, or other polynomial equation to solve.Solve the remaining equation. (use Unit 4 & 5 notes to help you)Check all solutions by plugging your answer into the original equation.Day 5: Inverse Functions Graphing Square & Cube Root FunctionsIn these notes we will learn What an inverse function is and how to find it algebraically. How to graph Square root and Cube root functions using transformations. Recall from Unit 2: An inverse function maps the output values back to their original input values. For the function: { (2, -2), (3, -3), (5, -5), (7, -7)} a. Create a mapping diagram for the function. b. Create a mapping diagram for the inverse function.Domain:Domain:Range: Range:What is the relationship between the domain/range of a function and the domain/range of its inverse?How could we tell a function and its inverse by the graph?f(x) = x2, x > 0 g(x) = 1793542429782.f(x) = 2x2 – 3, x > 0 g(x) = 22421152478Finding an Inverse Relation from an equation Switch x and y. Solve for y.Steps: Switch x and y in equation.Solve for y. 1. y = 4x + 2 2. f(x) = -2x + 5 3. f(x) = x2 + 2 , x 0 4. y = Verifying that Functions are Inverses of each other:1. Verify that f(x) = 4x + 2 and g(x) = are inverses of each other. Show that f(g(x)) = xANDShow that g(f(x)) = x2. Verify that the inverse functions you found above are correct:The Square Root Function: The parent function is Let’s look at its graph and table of values using our calculator: 323822333655xWhat is the domain of the square root parent function?Why?What is the inverse of the square root function? Sketch it. Why must we limit the inverse to one branch of the quadratic (parabola)?The Cube Root Function: The parent function is Let’s look at its graph and table of values using our calculator: x255079582550What is the domain of the square root parent function?Why?What is the inverse of the cube root function? Sketch it. Why can we use both braches of the cubic?Functions have the same transformations as the absolute value function y = a|x – h| + k.Given if : Vertically Stretch the graph by a factor of if : Vertically Shrink the graph by a factor of if : Reflect the graph about the x-axis(h, k): Translate the graph horizontally h units and vertically k units.Let c be a positive real number. Let .Vertical shift c units upward: Horizontal shift right c units: Vertical shift c units downward: Horizontal shift left c units: Transformations of Square Root Function: (h,k): _________642620-5080 (h,k): _________1119505180340 (h,k): _________798195170815 (h,k): _________1242638199191With SQUARE ROOT FUNCTIONS when you are completing the table of values…you will have x-values on ONE side of the initial point (h,k). Why?Transformations of Cube Root Function: (h,k): _________642620-5080 (h,k): _________1119505180340 (h,k): _________798195170815 (h,k): _________1242638199191Day 6: Graphing Square & Cube Root FunctionsIn these notes we will ANALYZE the graphs of Square Root and Cube Root FunctionsDomain Restrictions based on an equation:1. Dividing by zero is undefined: a denominator can NEVER be equal to zero.2. The square root of a negative number does not exist . . . we NEVER put a negative number under a square root (unless we are dealing in complex numbers). We will look at Case #1 in Unit 7. Case #2 above: No Negatives Under the Radical Sign!! x 0Do you have a square root? Do you have a rational power that has a denominator of 2? If not, then you don’t have to worry about this restriction.f(x) = f(x) =Domain: The set of all real numbers x ≥ 0Now let’s go back and define our characteristics from Unit 2 with the square root and cube root function. The Square Root Function: The parent function is (h, k): _____3890450109220x - Intercepts: ____________ y - intercept: ______________Domain: _____________________Range: _______________________Increasing: ____________________Decreasing: ___________________The Cube Root Function: The parent function is (h, k): _____3890450109220x - Intercepts: ____________ y - intercept: ______________Domain: _____________________Range: _______________________Increasing: ____________________Decreasing: ___________________Complete the following. Graph without a calculator. Then verify with your calculator and use to find your intercepts if necessary. Round to the nearest tenth. 1466850244475(h,k): __________x-int: ___________y-int: ___________Domain: _________Range: __________14363702540(h,k): __________x-int: ___________y-int: ___________Domain: _________Range: __________1466850162560(h,k): __________x-int: ___________y-int: ___________Domain: _________Range: __________143637080010(h,k): __________x-int: ___________y-int: ___________Domain: _________Range: __________ ................
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