) = nlog b ( m) ( m) log b ( ) ( ) = log a b ( ) Algebra 2 (1) Semester 2. Exponents and Logarithmic Functions

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1 Exponents and Logarithmic Functions Algebra 2 (1) Semester 2! a. Graph exponential growth functions!!!!!! [7.1]!! - y = ab x for b > 0!! - y = ab x h + k for b > 0!! - exponential growth models:! y = a( 1+ r) t!! - compound interest:! y = a 1+ r n! b. Graph exponential growth functions!!!!!! [7.2]!! - y = ab x for 0 < b < 1!! - y = ab x h + k for 0 < b < 1!! - exponential decay models:! y = a( 1 r) t! c. Functions involving e!!!!!!!! [7.3]!! - natural base e!! - graph natural base functions! y = ae rx! y = ae r( x h) + k!!! - growth if a > 0 and r > 0!!! - decay if a > 0 and r < 0!! - Continuously Compounded Interest:! A = Pe rt! d. Evaluating Logarithms and graph logarithmic functions!!! [7.4]!! - log b ( y) = x b x = y!! - domain and range of exponential and logarithmic functions!! - inverse graphs of exponential growth and decay!! - graph! y = log b ( x h) + k! e. Apply properties of Logarithms!!!!!! [7.5]!! - Product Property! log b ( mn) = log b ( m) + log b ( n) m!! - Quotient Property!! log b = log b n!! - Power Property!! log b ( m n ) = nlog b ( m)!! - Change-of-Base Formula! log c a nt ( m) log b n ( ) = log a b ( ) ( c)! f. Solve Exponential and Logarithmic Equations!!!! [7.6]!! - b x = b y if and only if x = y!! - log b ( x) = log b ( y) if and only if x = y ( x, y, b positive, b 1)!! - extraneous solutions log b ( )

2 Rational Functions! a. Model Direct, Inverse, and Joint Variation!!!!! [8.1]!! - Direct variation! y = ax!! - Inverse variation! y = a k!! - Joint variation! z = axy! or! p = aqrs! b. Graph Simple Rational Functions!!!!!! [8.2]!! - y = a x!! - y = a x h + k!! - y = ax + b cx + d!! - vertical and horizontal asymptotes, domain and range, holes! c. Graph General Rational Functions!!!!!! [8.3]!! - for f x ( ) = p ( x ) q( x)! 1. x-intercepts are the real zeros of p(x)!!!!! 2. vertical asymptote at each real zero of q(x)!!!!! 3. three cases for the horizontal asymptote!!!!!! - y = 0, ratio coefficients, or no asymptote! d. Multiply and Divide Rational Expressions!!!!! [8.4]!! - simplified form!! - factor 1!! - divide... multiply by reciprocal! e. Add and Subtract Rational Expressions!!!!! [8.5]!! - LCD!! - subtract... distribution of negative!! - complex fractions (2 methods)! f. Solve Rational Equations!!!!!!! [8.6]!! - cross multiplying!! - multiply each side by the LCD!! - extraneous solutions

3 Counting Methods and Probability! a. Apply the Counting Principle and Permutations!!!! [10.1]!! - tree diagram!! - fundamental counting principle!! - counting principle with repetition!! - permutation formula:! P = n! n r ( n r)!!! - permutation with repetition! b. Use Combinations and the Binomial Theorem!!!! [10.2]!! - combination formula:! C = n! n r ( n r)! i r!!! - card problems!! - multiply or add combinations!! - at least or at most!! - Pascalʼs triangle ( ) n!! - Binomial Expansions a + b!! - Find coefficient in an expansion! c. Define and Use Probability!!!!!!! [10.3]!! - theoretical probability!! - use combinations in probability!! - odds!! - experimental probability!! - geometric probability!!! d. Find Probabilities of Disjoint and Overlapping Events!!! [10.4]!!! - overlapping events:! P( A or B) = P( A) + P( B) P( A and B)!! - disjoint (or mutually exclusive):!! P( A or B) = P( A) + P( B)!! - complement of an event:!! P ( A ) = 1 P( A)! e. Find Probability of Independent and Dependent Events!!! [10.5]!! - Independent events:! P A and B ( ) = P( A) i P( B) ( )!! - Dependent events (conditional probability):! P( A and B) = P( A) i P B A

4 Sequence and Series! a. Define and Use Sequence and Series!!!!! [12.1]!! - sequence!! - terms!! - finite and infinite sequence!! - writing rules!! - series upper lim it!! - summation notation (sigma notation):! rule k=lower lim it! b. Analyze Arithmetic Sequences and Series!!!!! [12.2]!! - arithmetic sequence!! - common difference!! - nth term! a n = a 1 + (n 1)d!! - write a rule given two terms!! - arithmetic partial sum! S n = n a + a 1 n 2! c. Analyze Geometric Sequences and Series!!!!! [12.3]!! - geometric sequence!! - common ratio!! - nth term! a n = a 1 r n 1!! - write a rule given two terms 1 r n!! - geometric partial sum! S n = a 1 1 r! d. Find Sums of Infinite Geometric Series!!!!! [12.4]!! - sum of finite series for 1 < r < 1!! S = a 1 1 r! e. Use Recursive Rules with Sequence and Series!!!! [12.5]!! - explicit rule!!!!! - recursive rule! ( a n = a n 1 + d! or a n = r i a n 1 )!! - write rules!! - iterations

5 Complex Numbers! a. Perform Operations with Complex Numbers!!!!! [4.6]!!!! - imaginary unit i! i = 1!! - powers of i!! i 2 = 1! i 3 = i!! i 4 = 1!! - solve a quadratic equation with i!! - complex number:! a + bi!! - sum and differences of complex numbers!! - multiply complex numbers!! - complex conjugates!! - divide complex numbers!! - plot complex numbers in the complex plane!! - absolute value of a complex number