Unit 6: 10 3x 2. Semester 2 Final Review Name: Date: Advanced Algebra

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1 Semester Final Review Name: Date: Advanced Algebra Unit 6: # : Find the inverse of: 0 ) f ( ) = ) f ( ) Finding Inverses, Graphing Radical Functions, Simplifying Radical Epressions, & Solving Radical Equations =.) Reverse and y.) Solve for y.) Replace y with f ( ) ) f ( ) = ) f ( ) = 6 g If f ( ) = + and ( ) = +, find: ) f ( ) g( ) 6) g( ) f ( ) 7) Select ALL that apply. Which of the following has an inverse that is a function? A) B) C) *If a graph passes the vertical line test the graph itself is a function* D) E) F) If a graph passes the horizontal line test the inverse graph is a function.

2 8) f() = + + f ( ) = a h + k a = vertical stretch or shrink h = horizontal shift (left/right) opposite of sign k = vertical shift (up/down) same as sign Domain: Range: #9 : Write the following in radical form. 9) 0) ) eponent inde # 0: Simplify. ) 6 ) ) Negative eponents 8 move to opposite side of fraction (drop it like its hot!) ) 8 7 y 0 y z 6) y 6 yr 7) ab a b c Factor coefficient under the radical using factors that match the inde for eample: # use a perfect square, # use a power of and #6 use a perfect cube. For variables, determine how many times the inde goes into the eponent, leave remainders in the house 8) 0 9) 7 + 0) 7 Simplify radicals first then only combine like terms

3 # : Simplify: ) ) ) 6 Similar to: + 7 (When multiplying bases, add eponents) # 9: Solve, be sure to check for etraneous solutions. ) 9 b = b 7 ) 7b 0 = b 6) k = k Square both sides to cancel radical, you must rewrite the binomial twice and distribute. Checking for etraneous solutions: Sub answer back in and make sure both sides are equal 7) = + ( ) 6 8) = ( ) 9) ( ) 9 = Isolate the quantity with the rational eponent. Raise both sides to the power of the inde (denominator). Then solve for. Check for etraneous solutions: When subbing in you should get a true statement. Eponential and Logarithmic Functions, Epressions, and Equations 0) Which of the following represents an equation of Eponential Growth? A. y = Unit 7: B. y (. ) = 08 C. y = D. y = f ( ) = a b Growth: b > Decay:

4 ) Which of the following represents an equation of Eponential Decay? A. y = B. 8 ( ) y =. C. y = D. y # 7: Solve. Round to the nearest thousandth when necessary. = b < ) 6 p = 6 ) = ) b b 7 = + When possible, change the larger base into the smaller base (think of your eponent chart) ) n+ 0 = 6) + = 7) 0n = If changing to same base is not possible, take the log of both sides, which makes the eponent become the coefficient, then finish solving for the missing variable. #8 0: Rewrite the equation in logarithmic form. 8) 8 = 9) = 8 0) = 0 The base is also the base on the log, the number the eponential epression equals becomes the argument, and the new logarithmic epression equals the eponent. E: b a loga c = c = b # : Evaluate. ) log ) log6 ) log66 8.) Set the epression equal to..) Switch to eponential form..) Change to same base..) Solve by setting the eponents equal to each other.

5 # 9: Solve. Be sure to check for etraneous solutions. Round to the nearest thousandth. ) log = ) log ( ) = 6) ( ) 8 log 7 6 =.) Switch to eponential form..) Simplify the base and eponent..) Solve for. 7) log ( 8 + 7) = log8 ( ) 8) log ( ) ( + = log + ) When both sides are logs, set the arguments equal to each other and solve. 9) log ( ) = log ( ) #0 : Epand. 0) log ab c ) ( ) log6 y ) log y Multiplication to Addition Division to Subtraction Eponents become the Coefficient # : Condense. ) log8 + log8y ) log7u + log7v log7w log7z Work backwards from # -. There should only be one log in your answer! ) log a + log b log a log c

6 #6 8: Identify the domain and range ) f ( ) = 7) f ( ) = + 7 8) f ( ) 6 = + 8 h ( ) f = a b + k When a > 0:, Domain: ( ) Range: ( k, ) Domain: Domain: Domain: Range: Range: Range: #9 6: Determine the end behavior for the following graphs: When a < 0:, Domain: ( ) Range: (,k) 9) 60) 6) ( ) As,f? ( ) As,f? t #6 6: Use the general model At ( ) a( r) = + to determine the following. 6) In 00, there were. million Facebook members. Each year Facebook grows by %. In what year does the Facebook reach 0 million users? When the eponent is the unknown, take the log of both sides. 6) In 000 you deposited $00 in an account that pays % annual interest. How long does it take the balance to reach $000?

7 Unit 8: Trigonometric Ratios, Coterminal Angles, Radians to Degree Conversion, Reference Angles, Unit Circle and Graphing Trig Functions 6) If sin θ= and θ is an acute angle of a right triangle in Quadrant I, then tan θ=? 8 6) If cos θ=, and θ is an acute angle of a right triangle in Quadrant I, then tan θ=? S C T O H A H O A #66 67: Use the figure to the right to find the measure of the missing angle. 66) Measure of B 67) Measure of A B Inverse trig to find the measure of angles A C 68) An airplane flying at an altitude of 8,000 feet spots an island. The angles of depression of the opposite ends of the island are º and º. What is the width of the island? 69) Walt and Julia are on the opposite sides of a tower of 60 meters height. They measure the angle of elevation of the top of the tower as 0 and respectively. Find the distance between Walt and Julia.

8 #70 7: List at least three angles that are coterminal with the following: 70) 0 7) π Add or subtract 60 Radian measures: Add or subtract π #7 7: Epress the following using the equivalent measure in degrees or radians. 7) π 7) 7 9 To convert to degrees: Multiply by 80 π To convert to radians: Multiply by π 80 #7 7: Determine the reference angle. 7) π 7) π 7 Sketch the angle Determine the measure of the angle between the terminal side and the ais #76 77: List the quadrants for θ. 76) If sin θ< 0 77) If cos θ> 0 (,y) ( cosθ, sinθ) #78 8: Find the eact value. π 78) sin 79) π cos Use the unit circle and the tall, middle, short triangles!

9 7π 80) sec 6 8) 7π cos #8 8: Find the value for θ. 8) cotθ = 8) cosθ = 8) sinθ = There may be multiple answers since can be negative or positive in two quadrants. #8 86: Solve the equation for θ. 8) cosθ = 0. if 70 < θ < ) sinθ = 0. if 90 < θ < 80.) Find the reference angle by using an inverse.) Sketch the reference angle in the stated quadrant.) Determine the measure of the entire angle. #87 88: Determine the interval for which the function is increasing and decreasing from[ 0, π ], write at least two equations for each and find the period. 87) 88) Increasing: Increasing: Period: Cosine: Ma, 0, Min, 0, Ma Sine: 0, Ma, 0, Min, 0 Decreasing: Decreasing: Period: Period:

10 Identify the key features associated with this function: 89. y = sin ( + π ) 90. y = cos ( π 8 ) + Amplitude: Vertical Shift: Period: Phase Shift: Amplitude: Vertical Shift: Period: Phase Shift: Unit 9: Rational Epressions, Rational Equations, Vertical and Horizontal Asymptotes #9 9: Determine where the function is undefined. Undefined is when 6 the denominator 9) y = 9) y = 9) y = 9 8 equals 0. #9 96: Factor: 9) 6 9) ) Perfect Cubes: + + ( )( ) ( + )( + ) Cube roots Squares Perfect Squares: ( )( + ) #97 0: Simplify: 97) 0 p p + 8p 8 98) r + r + r 8 r + 99) One epression: Factor, Cancel Multiplying: Factor, Cancel Dividing: Factor, Flip, Cancel

11 00) ) + 0) Adding:.) Find a common denominator.) multiply by whatever is missing in each.) Combine like terms.) Try to factor numerator in order to simplify #0 0: Solve. Be sure to check for etraneous solutions. 0) = 0) 9 = ) 6 + = 9 8.) Determine the common denominator.) Multiply every fraction by the common denominator.) Cancel the fractions.) Solve #06 08: Determine the vertical asymptotes ) y = 07) y = + 08) y 8 8 = + + Vertical Asymptotes: Same way as finding where the function is undefined (set the denominator = 0) #09 : Determine the horizontal asymptotes ) y = 0) y = + ) y 8 8 = + + Horizontal Asymptotes: Divide like coefficients (If a term is missing insert a zero to hold its place!!) E: y = y = 0 + H.A. y = 0

12 Unit 0: Sequences and Series: Finding the First Terms, Writing the rule, Finding the Sum # : Find the first five terms of the following sequences. Then state the term that is asked. n + ) a n = ) a k = a k +, a = 8 n Eplicit: Sub in n Recursive: Start with the given first time and then use the rule to find the net. *state the rd term: *State the th term: # : Find a n for the arithmetic sequence given the following terms: ) a = anda = ) a 0 = 8 and a 6 = 0 Arithmetic: a = dn + c n #6 7: Find a n for the geometric sequence given the following terms: 6) a = 6 and a 6 = 97 7) a = 6 and a = 96 Geometric: n a = a r n

13 #8 : Find the sum: 8) k 9) k = k = 8 k 0) k = 0k 0 SUM of Arithmetic (finite): n Sn = ( a + an) SUM of Arithmetic (infinite): No Sum k k ) ( ) ) ( ) ) ( ) k = k = k = k SUM of Geometric (finite): ( n a r ) sn = r SUM of Geometric (infinite converging r < ): a S = r SUM of Geometric (infinite diverging r >): No Sum