ALGEBRA 2/TRIGONMETRY TOPIC REVIEW QUARTER 2 POWERS OF I

Transcription

1 ALGEBRA /TRIGONMETRY TOPIC REVIEW QUARTER Imaginary Unit: i = i i i i 0 = = i = = i Imaginary numbers appear when you have a negative number under a radical. POWERS OF I Higher powers if i: If you have an imaginary term with a power on it higher than, you must divide the power by and replace the old power with one indicated by the remainder when you divided. Remainders: Ex) i =.5 => 0 =.00 = i 0 i 59 = i = -i =.5 = i =.50 = i =.5 = i ) If 8 is subtracted from 6, the difference is () i () 5 () i () i ) When simplified, i + i i + i is equal to () i () () i () + ) If f(x) = x, what is the value of f ( i)? () 8 () - () - ()

2 MULTIPLYING/DIVIDING/GRAPHING COMPLEX NUMBERS Complex Number: made up of a real # and imaginary # a + bi Real # Imaginary # Adding and Subtracting Complex Numbers Combine like terms: (5 + i) + ( - 5i) = i Multiplying Complex Numbers First remember that i = and that i = - FOIL the binomials Simplify the expression into a + bi form Dividing Complex Numbers: Dividing complex number is the same as rationalizing with radicals. We also do not like imaginaries in the bottom of a fraction. So:. Monomial in the bottom: Multiply the top and bottom of the fraction by only the imaginary term in the denominator. Simplify.. Binomial in the bottom: Multiply the top and bottom by the conjugate of the bottom (same binomial with diff. sign between). Simplify. Graphing Complex Numbers: Graphing a complex number is like graphing a coordinate except instead of (x,y), you graph (a,b) from a + bi form. You can graph it as a point or as a vector connecting the origin and that point. ) Expressed in a + bi form, ( + i) is equivalent to () 5 + i () 5 () + i () 5) Expressed in a + bi form, 5 + i is equivalent to () i () + i () + i () i ) In which quadrant does the sum of + i and 5i lie? () I () II () III () IV

3 SOLVING QUADRATICS WITH COMPLEX NUMBERS/DISCRIMINANT DISCRIMINANT: b ac Used to describe the nature of the roots If: b ac > 0 and non-perfect square roots are real, irrational, and unequal b ac > 0 and perfect square roots are real, rational, and unequal b ac = 0 roots are real, rational, and equal b ac < 0 roots are imaginary Solving Quadratics with complex numbers: The process for solving all quadratics always remains the same. You should:. Try to factor first. If you can t factor, then try. Completing the Square. Quadratic Formula You will get imaginary roots if you end up taking the square root of a negative discriminant. Things to keep in mind: roots are where the graph crosses the x-axis if your roots are imaginary, the graph will not cross the x-axis. if your roots are =, touches x-axis once (tangent) if your roots are unequal, graph crosses x-axis twice (goes UNDER the x-axis line) 5 ) What is the solution set of the equation x + =? x () ± i () ± i () ± i () ± i 8) The roots of the equation x + x = 0 are () real, rational, and unequal () real, rational, and equal () real, irrational, and unequal () imaginary 9) Which of the following graphs would best represent a parabola having a discriminant equal to 0? () () () ()

4 SEQUENCES AND SERIES Explicit Formula: Plug value of n in to get any term. Ex: Given: cn = -n +, the 5 th term would be c5 = -5 + = Recursive Formula: need the term that came before. Ex: Given: j = () = 9 j = (9) = 8 j =, the j = ( j ) rd term would be n n Finding the Common Difference: Subtract the two terms you know and set equal to the number of spaces between the two numbers times d. Ex. Given: 5 80, find the comm. diff 80 5 = 5 5 = 5d d = 5 Finding the Common Ratio: Divide the two terms you know and set equal to r raised to the number of spaces between the two numbers. Ex. Given: 8, find the comm. ratio 8 = 8 = r r = 8 Arithmetic Sequence (addition): an = a + (n )d Where: a = st term, n = term in question d = common difference (constant you add each time) Geometric Sequence (multiplication): n g n = g ( r) Where: g = st term, n = term in question r = common ratio (constant you mult each time) Arithmetic Series (addition): n(a +a ) n Sn = Where: a = st term, n = term in question, an = last term Geometric Series (multiplication): g ( r n ) Sn = r Where: g = st term, n = term in question, r = common ratio 0) Which of the following sequences is arithmetic? (),,, 8 (),,, (),,, 5 (),, 9, ) Find the common ratio for the geometric sequence for which t = 6 and () () - () t () = ) Find the fifth term of the sequences, 6, -, () () -8 () - () 8 ) What is the sum of the series ? (), 00 (), 95 (), 90 (),00

5 SUMMATION By Hand: Find each term and then add the terms up: Upper Bound Lower Bound Formula On Calculator: Buttons are in the LIST menu SUM (SEQ (formula, X, lower bound, upper bound)) Note: If there is any number in front, follow the operation. 9 (n ) + 0 (m + ) n = 5 m = Mult sum by Add 0 to the sum ) What is the value of k= ( k + )? () () () 5 () 0 5) How is the series written sigma notation? 00 () n= 00 n () n= 99 n () n= 99 n () n= n + 6) What is the value of x= (cos xπ) () () - () 0 ()

6 LAW OF EXPONENTS If the bases are the same, when Multiplying Keep the base the same and ADD the exponents. Dividing Keep the base the same and SUBTRACT/CANCEL the exponents Power Law- If a base with an exponent is raised to another exponent, MULTIPLY the exponents If there are multiple bases raised to an exponent be sure to distribute the exponent to ALL the bases. Zero Exponents (anything) 0 = Negative Exponents CONVERTED TO Positive Exponents. Find base with negative exponent. Relocate base, if base is in the numerator of a fraction relocate the base to the denominator -- change neg exponent to a positive exp. denominator of a fraction relocate the base to the numerator -- change neg exponent to a pos exp. Fractional Exponents converted to Radical Form ( 5x) ( 5x) numerator of the fraction stays as the exponent to the base. denominator of the fraction goes in the crack of the radical. (root of the radical) ) The expression ( x y ) is equivalent to () 9x 6 y 9 () x 5 y 6 () x 6 y 9 () x 5 y 6 8) If b > 0, the expression b () ( b ) () is equivalent to ( b) () ( b) () ( b) 9) The expression 8t t 5 s s, where t 0 and s 0 is equivalent to () t s () t s () s t () t s

7 EXPONENT EQUATIONS Solving equations with fractional exponents: Isolate base with fractional exponent Raise each side to the reciprocal power o If reciprocal power has an EVEN NUMBER IN THE DENOMINATOR there w will be answers. ( x + ) = 6 (( x + ) ) = 6 x + = ± (6) x + = ± 8 x = 0 and x = 6 Solving exponential equations with same bases: Cross bases out Set exponents = to each other and solve Eponential growth/decay equations: Start with y = ab x, o a = initial (starting) value) o b = growth rate Increasing (+ %decimal) Decreasing ( %decimal) o x = time frame Solve for y type in calculator. Solve for x isolate ( ), then change into log form. Compounding Interest: n times a year, use: A = P + r n continuously, use: A = Pe rt (nt) Solving exponential equation with different bases: Get bases to be the same number raised to a different exponent. o o Cross bases out Set exponents = to each other and solve 0) Solve: ( c ) = 5 () -6 and () 6 and - () - () 6 ) Solve: 6 x+ = 8 x () 8 x = () x = () 8 8 x = () x = 8

8 ) Kathy deposits $5 into an investment account with an annual rate of 5%, compounded annually. The amount in her account can be determined by the formula A = P( + R), where P is the amount deposited, R is the annual interest rate, and t is the number of years the money is invested. If she makes no other deposits or withdrawals, how much money will be in her account at the end of 5 years? () $.5 ()$9.9 ()$5.5 () $5.9 t ) Rhonda s car is expected to depreciate at a rate of 0% per year. If her car is currently valued at $8,000, to the nearest dollar, how much will it be worth in five years? () $,69 ()$,00 ()$9,5 () $5,600 ) The strength of a medication over time is represented by the equation x y = 00(.5), where x represents the number of hours since the medication was taken and y represents the number of micrograms per millimeter left in the blood. Which graph best represents this relationship? () () () () 5) What is the asymptote of the graph x f (x) = e? () x = 0 () y = 0 () y = () y = x 6) To slide the graph of the equation equation? () x x y = two units left, the equation must be altered. What is the new x y = () y = () y = () y = x + x +