graphing square root functions - njctlcontent.njctl.org/courses/math/algebra-ii/radical...graphing...
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Algebra II
Radical Equations
www.njctl.org
2015-04-21
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Roots and RadicalsTable of Contents:
Working with Square Roots
Cube Rootsnth Roots
Irrational Roots
Solving Radical EquationsRational Exponents
Rationalizing the Denominator
Adding and Subtracting Radicals
Graphing Square Root Functions
Complex Numbers
Multiplying Radicals
click on the topic to go to that section
If review is needed before or during this unit click on the link below.
Fundamental Skills of Algebra (Supplemental Review)
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Graphing Square Root Functions
Return to Table of Contents
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The inverse of is
Recall the Inverse of Squares...
...but the result is not a function.
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The domain of y = x2 is restricted to x ≥ 0, so that the inverse will be a function.
Domain: [0, # )Range: [0, # )
Domain: [0, # )Range: [0, # )
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3-3
The function is one of the parent functions. Use this fact to help you anticipate the graph or find the function from the graph.
Remember transformations...
Domain: [0, # )Range: [0, # ) Domain: [-3, # )
Range: [0, # )
Domain: [3, # )Range: [0, # )
Parent
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-3
3
And...
Domain: [0, # )Range: [0, # )
Domain: [0, # )Range: [3, # )
Domain: [0, # )Range: [-3, # )
Parent
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1 Which is the graph of the function?
A B
C D
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2 Which is the graph of the function?
A B
C D
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3
3 Which is the equation of the graph?
A
B
C
DTe
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Slide 17 / 272Remember what happens when you have af(x) or f(bx)...
Domain: [0, # )Range: [0, # )
Domain: [0, # )Range: (-# , 0]
Domain: [0, # )Range: [0, # )
Parent
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And...
Domain: (-# , 0]Range: (-# , 0]
Domain: (-# , 0]Range: [0, # )
Domain: [0, # )Range: [0, # )
Parent
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8 Which is the graph of the function?A B
C D
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9 Which is the graph of the function?A B
C D
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10 Which is the equation of the graph?
A
B
C
D
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11 Which is the equation of the graph?
A
B
C
D
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What happens when you combine the transformations? Why is this one only moved to the right 2?
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In order to see how much the graph moves horizontally, any coefficient must be factored out. Rewrite the following by factoring out the coefficient:
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Slide 31 / 272 Slide 32 / 272Graph the function:
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1. Anticipate the graph using transformations.2. Set the radicand ≥ 0 and solve to find the domain.3. Choose 3 values in the domain that will give you perfect squares under the radical. Use a t-table to find y's.4. Plot the points and graph.
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Graph the function:Te
ache
r1. Anticipate the graph using transformations.2. Set the radicand ≥ 0 and solve to find the domain.3. Choose 3 values in the domain that will give you perfect squares under the radical. Use a t-table to find y's.4. Plot the points and graph.
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Graph the function:
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1. Anticipate the graph using transformations.2. Set the radicand ≥ 0 and solve to find the domain.3. Choose 3 values in the domain that will give you perfect squares under the radical. Use a t-table to find y's.4. Plot the points and graph.
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Working with Square Roots
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of Contents
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17 What is ?Te
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18 Find:
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19 What is ?
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20 What is ?
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What happens when you have variables in the radicand? To take the square root of a variable rewrite
its exponent as the square of a power.
Variables
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IMPORTANT: When taking the square root of variables, remember that answers must be positive. Even powered answers, like the last page, will be positive even if the variables are negative. The same cannot be said if the answer has an odd power. When you take a square root and the answer has an odd power, put the result inside an absolute value symbol.
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27
A
B
C
D no real solution
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28
A
B
C
D no real solution
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29
A
B
C
D no real solutionTe
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30
A
B
C
D no real solution
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Return to Table
of Contents
Irrational Roots
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Simplifying Radicals
is said to be a rational number because there is a perfect square that equals the radicand.
If a radicand cannot be made into a perfect square, the root is said to be irrational, like .
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The commonly accepted form of a radical is called simplest radical form.
To simplify numbers that are not perfect squares, start by breaking the radicand into factors and then breaking the factors into factors and so on until only prime numbers are left. This is called prime factorization.
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31 Which of the following is the prime factorization of 24?
A 3(8)
B 4(6)
C 2(2)(2)(3)
D 2(2)(2)(3)(3)Te
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32 Which of the following is the prime factorization of 72?
A 9(8)
B 2(2)(2)(2)(6)
C 2(2)(2)(3)
D 2(2)(2)(3)(3)
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33 Which of the following is the prime factorization of 12?
A 3(4)
B 2(6)
C 2(2)(2)(3)
D 2(2)(3)
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34 Which of the following is the prime factorization of 24 rewritten as powers of factors?
A
B
C
D
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35 Which of the following is the prime factorization of 72 rewritten as powers of factors?
A
B
C
D
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36 Simplify:
A
B
C
D already in simplified formTe
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37 Put in simplest radical form:
A
B
C
D already in simplified form
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38 Put in simplest radical form:
A
B
C
D already in simplified form
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39 Simplify:
A
B
C
D already in simplified form
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40 Which of the following is not an irrational number?
A
B
C
D
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42 Simplify:
A
B
C
D
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44 Put in simplest radical form:
A
B
C
D
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The same process goes for variables, but absolute value signs need to be included where appropriate.
Absolute value symbols are required when the initial exponent is even and the exponent after taking the root is odd. If the initial exponent is odd, you will not need absolute values.
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Examples:
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46 Simplify:
A
B
C
D
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47 Put in simplest radical form:
A
B
C
D
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48 Simplify:
A
B
C
D
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49 Put in simplest radical form:
A
B
C
D
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50 Put in simplest radical form:
A
B
C
D
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Adding and Subtracting Radicals
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Adding and Subtracting Radicals*Note: When adding or subtracting radicals, you do not add or subtract the radicands (the inside).
Consider:
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To add and subtract radicals they must be like terms.
Radicals are like terms if they have the same radicands and the same indexes.
Like Terms Unlike Terms
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51 Identify all of the pairs of like terms:A
B
C
D
E
F
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53 Simplify:
A
B
C
D Already Simplified
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57 Simplify:
A
B
C
D Already in simplest form
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60 Simplify:
A
B
C
D Already in simplest formTe
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Multiplying Radicals
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Whole number times whole number and radical times radical. Never multiply a whole number and radical! Leave
all answers in simplest radical form.
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Examples:
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Examples:
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63 Multiply:
A
B
C
D
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64 Simplify:
A
B
C
D
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65 Simplify:
A
B
C
D
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66 Simplify:
A
B
C
D
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67 Simplify:
A
B
C
DTe
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Multiplying Polynomials with RadicalsLeave all answers in simplest radical form
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68 Multiply and write in simplest form:
A
B
C
D
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69 Multiply and write in simplest form:
A
B
C
D
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70 Multiply and write in simplest form:
A
B
C
D
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71 Multiply and write in simplest form:
A
B
C
D
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72 Multiply and write in simplest form:
A
B
C
DTe
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Rationalizing the Denominator
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Mathematicians don't like radicals in the denominators of fractions.
When there is one, the denominator is said to be irrational. The method used to rid the denominator is termed "rationalizing the denominator".
Which of these has a rational denominator?
RationalDenominator
IrrationalDenominator
Rationalizing the Denominator
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If the denominator is a monomial, to rationalize, just multiply top and bottom of the fraction by the root part of the denominator.
Examples:
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Multiplying by the conjugate turns an irrational number into a rational number.
Check out what happens... Teac
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Do you see a pattern that let's us go from line 1 to line 3 directly?Example Example Example
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Use conjugates to rationalize the denominators:
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73 What is conjugate of ?
A
B
C
D
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75 Simplify:
A
B
C
D Already simplified
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76 Simplify:
A
B
C
D Already simplified
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77 Simplify:
A
B
C
D Already simplified
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80 Simplify:
A
B
C
D
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Already simplified
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Cube Roots
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If a square root cancels a square, what cancels a cube?
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Try...
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Put in simplest radical form:
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87 Simplify:
A B
C D not possible
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88 Simplify:
A B
C D not possible
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89 Simplify:
A B
C D not possible
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90 Simplify:
A B
C D not possible
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91 Simplify:
A B
C D not possibleTe
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92 Put in simplest radical form:
A
B
C
D not possible
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nth Roots
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Try...
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95 Simplify:
A
B
C
D
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96 Simplify:
A
B
C
D
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98 Simplify:
A
B
C
D
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99 Simplify:
A
B
C
D
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100 Simplify:
A
B
C
DTe
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Try...Te
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103 Simplify:
A
B
C
D
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104 Simplify:
A
B
C
D
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Remember that , given an nth root in the denominator, it will need to be rationalized. To rationalize, find the complement if the nth root that will create a perfect root in the denominator. Multiply top and bottom by the complement. Simplify.
Examples:
Rationalizing nth roots of monomialsTe
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Try:
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106 Rationalize:
A
B
C
D
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108 Rationalize:
A
B
CD
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112 Simplify:
A
B
C
D
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Rational Exponents
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Rational exponents, or exponents that are fractions, are another way to write and work with radicals.
Power
Root
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Simplify:Te
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115 Simplify:
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116 Simplify:
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117 Simplify:
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Rewrite each radical as a rational exponent in the lowest terms.
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Rewrite each expression as a single radical. To combine more than one number or variable, the roots must be the same. Te
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When the roots (denominators) are different, they must be made into a common number in order to create a single root.
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120 Find the simplified expression that is equivalent to:
A
B
C
D
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121 Find the simplified expression that is equivalent to:
A
B
C
D
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122 Find the simplified expression that is equivalent to:
A
B
C
D
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123 Simplify:
A
B
C
D
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124 Write with rational exponents:
A
B
C
D
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125 Find the simplified expression that is equivalent to:
A
B
C
D
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126 Write the following with exponents:
A
B
C
D
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Just like other problems where you must rationalize denominators, mathematicians like to have a an integer power in the denominators.
Therefore, if there is a fractional exponent in the denominator after simplifying, rationalize the denominator. Te
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128 Simplify:
A
B
C
D
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130 Simplify:
A
B
C
D
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Solving RadicalEquations
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Example:Te
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133 Find the solution to:
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134 Find the solution to:
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135 Find the solution to:
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137 Find the solution to:Te
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138 Solve the following:
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140 Solve:
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Complex Numbers
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Rational Numbers
Irrational Numbers
Imaginary Numbers
Real Numbers
Whole Numbers
Natural Numbers
Complex Numbers: All numbers are technically considered complex numbers. Real Numbers can be written as a + 0i - no imaginary component.
Integers
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Complex Numbers
Why does this work?
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Higher order i's can be simplified into i, -1, -i, or 1.
If the power of i is even: If the power of i is odd:...and the exponent is a multiple of 4, then it simplifies to 1.
...and the exponent is a multiple of 2,but not 4, then it simplifies to -1.
...factor out one i to create an even exponent. Use the rules for even exponents and leave the factored i.
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142 Simplify:
A i
B -1
C -i
D 1
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143 Simplify:
A i
B -1
C -i
D 1
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144 Simplify:
A i
B -1
C -i
D 1Te
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145 Simplify:
A i
B -1
C -i
D 1
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Simplify radical expressions that have a negative by taking out i first. Then, perform the indicated operation(s). Simplify any expression that has a power of i greater than one.
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147 Simplify:
A
B
C
D
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150 Simplify:
A
B
C
D
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Working with Complex Numbers
Operations, such as addition, subtraction, multiplication and division, can be done with i.
Treat i like any other variable, except at the end make sure i is at most to the first power.
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When multiplying, multiply numbers, multiply i's and simplify any i with a power greater than one.
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Multiply and leave answers in standard form.
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151 Simplify:
A
B
C
D
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152 Simplify:
A
B
C
D
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153 Simplify:
A
B
C
DTe
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154 Simplify:
A
B
C
D
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155 Simplify:
A
B
C
D
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Dividing with iSince i represents a square root, a fraction is not in simplified form if
there is an i in the denominator. And, similar to roots, if the denominator is a monomial just multiply top and bottom of the fraction
by i to rationalize.
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156 Simplify:
A B
C DTe
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157 Simplify:
A B
C D
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160 Simplify:
A
B
C
DTe
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161 Simplify:
A
B
C
D
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162 Simplify:
A
B
C
D
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