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Algebra 2 Spring Semester ReviewName From your learning this semester, you should be able to:Graph cubic, cube root/square root, rational, exponential and logarithmic functions using transformations – identifying features of the graphs such as vertical asymptotes, horizontal asymptotes, domain and rangeAdd/subtract/multiply polynomial expressionsFactor polynomials of degree 3 or 4 – including sum/difference of cubes, difference of squares, and greatest common factorDivide polynomials – using long division or synthetic divisionFind the constant of an inverse variation problemWrite equations of rational functions from graphs or given informationAdd/subtract/multiply/divide rational expressions Solve rational equations – including work rate; distance, rate, time problems; with proportions; including the possibility of extraneous solutionsFind the inverse of a cube root/square rootTranslate between rational exponents and radical expressionSimplify expressions with rational exponents using properties of exponentsSolve radical equations - cube root/square rootDetermine the recursive and explicit rule for a geometric sequenceWrite an exponential growth/decay equation from a situation or a graphCreate/select an appropriate regression model when given data – exponential, quadratic or linearMake a prediction from an exponential modelConvert between exponential and log formSimplify logarithms using the power propertyEvaluate log models when given input valuesSolve exponential equations using a common base and logarithmsSolve logarithmic equations – with possibility of extraneous solutionAlgebra 2 Spring Semester ReviewName___________________________Directions: Work each of the following problems on a separate sheet of paper and show all work. A graphing calculator is allowed.a. Justin drove his pickup truck about 22,000 miles in 2010. He read that in 2004, the average residential vehicle traveled about 10,200 miles, which increased by about 2.9% per year. Write a function for the average miles, m, as a function of time, t, the years since 2004. b. In 2010, did Justin drive more or fewer miles than the average driver? c. Justin bought his new truck for $32,000. Its value decreases 9.0% each year. Write a function for the value of the truck, v, as a function of time, t. d. When will the value of Justin’s truck fall below half of what he paid for it?For a pendulum with a length, L, in meters, the expression T=2πLg models the time in seconds for the pendulum to complete one swing. To find the length of the pendulum, solve this equation for L. If a = x6, what is 4a ? If 312x+28=4, what is the value of x3? Perform the indicated operation and simplify: x-1x2+3x+2+xx+1 Solve: 15nn-3= 5n-3-8 7. Solve: xx-3+x2=6x2x-6 Julien can mulch a garden in 20 minutes. Together, Julien and Remy can mulch the same garden in 11 minutes. How long will it take Remy to mulch the garden when working alone? Write logx(164) = -3 in exponential form. Use mental math to solve for x. Perform the indicated operation and simplify: x5-4x3x2-x-2x5-x4-2x3x2-1 A river barge travels at an average of 8 mi/hr in still water. The barge travels 60 mi up the Mississippi River and 60 mi down the river in a total of 16.5 h. What is the average speed of the current in this section of the Mississippi River? (Round to tenths) If y is inversely proportional to x, find the constant of variation when x is 8 and y is 4. Also, estimate the value of x when y = 16. 27772417556500Graph the equation fx=-(x-3)3+4 4216400-17716500Given this graph, draw the inverse. What makes these two graphs inverses of each other? Divide using synthetic division. (x4+3x2-5x-20) ÷(x-2) A rectangle has a width equal to x2+2x and a length equal to x-5. What is the perimeter?Add or subtract to simplify the expression.x5+3x2+1+(2x3-x5+4)b) 2x4+4x3+x-2-(x4+2x3+5x2-8)Given f(x) = log (x), write a verbal description using transformations of f(x) = -2log(x + 1) – 3 and write the domain and range in all three notations.Multiply or divide to simplify the expression. (x2+2x-1)(x3-5x+3)b) (2x3+4x2-7x+1)÷(x2-5)Perform the indicated operation and simplify: 2x2-48x2-16 - x+6x+4 Use the power property of logarithms to solve. log3(2x-3)12=1459041512763500Given 2logx4=16. Solve for x. Graph the function and give the vertical and horizontal asymptotes. y=21x-2+4.Train A is 15 miles per hour faster than Train B whose speed is x. Train A travels 430 miles in the same time that Train B travels 355 miles. Set-up and equation to find the speed of both trains and then find both speeds. Factor the polynomial.a) x3-2x2-4x+8b) 27x3+1 c) 8x3-644049164738900If the parent function f(x) = log x has been vertically stretchedby a scale factor of 4, shifted 1 unit right and 3 units down, what is the equation of the asymptote. Graph the function. Find the inverse of the following equations; make sure to give a domain restriction or a ± when needed. fx=3x-5 b.fx=x-22-4 c. fx=73x+1 d. fx=2x+13283463913492300Write the equation in the form of y=a1x-h+k for the following graph where the point (5, 0) is on the graph. Solve x+1812=x-2 Find the solution to the equation: 2x+4=4 Solve: 8x=2x+632. Solve: 14x=8x-1 33. Solve: 2?3x-1=162Rewrite in logarithmic form.e4=x b. 25=3x219011516510000Graph the square root function. y=3x+6-4Solve: 3x-33=1 Solve. logx2-18=log(-7x)Write the equation of a rational function that has a vertical asymptote at x = 4, a horizontal asymptote at y = 1, and contains the point (-3,0). Perform the indicated operation and identify any x-values for which the expression is undefined: 2x2-x-20+3x2+7x+1236093401460500 Describe the transformation from the cubed root parent function and then graph the functiony=3x-5+4Simplify: a) 8a613 b) 493243 c) x14x12 ? x34 Write the explicit and recursive rule for the geometric sequence.28, 14, 7, … Solve the rational equations. List the values of x that make the function undefined.xx+2=1xb) xx-6+5x+3=-45x2-3x-18Tell whether the function shows exponential growth or decay and give the rate of increase or decrease as a percent: fx=100(0.9)x b) fx=13(1.3)x c) fx=0.434x Write in exponential form.a) log2x+1=4 b) log(x-2)16=2 The time, t, it takes for a group of volunteers to construct a house varies inversely as the number of volunteers, v. If 20 volunteers can build a house in 62.5 working hours, how many volunteers would be needed to build a house in 50 working hours4429125-29527500Graph the functionfx= 2x+3-2 and state the domain and range.Multiply.x2-4x-5x2-3x+2?x2-4x2-3x-10On federal income tax returns, self-employed people can depreciate the value of business equipment. Suppose a computer valued at $2765 depreciates at a rate of 30% per year. Estimate the number of years it will take for the computer’s value to be less than $350. 218661269240A.B.C.A.B.C.Look at the graphs, what type of model would be appropriate for each scatterplot? (exponential, quadratic, or linear) A new medication is being studied to see how quickly it is being metabolized in the body. The table shows how much of an initial dose of 15 mgs remains in the bloodstream after different intervals of time. Use your graphing calculator to find the exponential regression model for this situation.How much of the drug remains in your bloodstream after 12 hours?Evaluate the logarithm fx=log2x-3f(1)b) f(11)c) f(35)Algebra 2 Spring Semester Review - KEYa. m(t) = 10,200(1 + 0.029)tb. More--about 9900 milesc. V = 32,000(1- .09)td. about 8 yearsgT24π2=L x32 or x327x2+3x-1(x+2)(x+1)orx2+3x-1x2+3x+2 n=2923x = 0, x = 7 24 minutesx-3 = 164, x = 4(x+2)(x-1)(x-2)(x+1)orx2+x-2x2-x-2 2.4 mphx = 2 and k=322419072603500 The graphs are reflections over the line y=x. This means that the domain of one graph is the range of the other graph (the x- and y-values have changed places).12858754318000-381004318000 (x-2)(x3+2x2+7x+9)-2 2x2 + 6x-10 a) 2x3+3x2+5 b) x4+2x3-5x2+x+6 The graph has been reflected across the x-axis and vertically stretched by a factor of 2. It has been shifted 1 unit left and 3 units down.a)x4-3x3-8x2+11x-3x2-52x+4+3x+21x-6x-4x=619030122476500 x = 100Vertical asymptote x=2 Horizontal asymptote y=443015+x= 355xTrain A – 86 mph; Train B – 71 mph a) x+2x-22b) 3x+19x2-3x+1190433776310008(x-2)(x2+2x+4) x=1a) f-1x=x+532;{x|x≥-5} y-1=2±x+4f-1x=x73-1f-1x=3x-12y=-21x-3+1x = 7 (x = -2 is extraneous) x=6 x=3x=35 x = 5a. lnx=4 b. log325=x Square Root: 2419353365500 36. x = 30 x=-9;x≠2, extraneous y=7(x-4)+1 or y=71(x-4)+15x-9(x+3)(x+4)(x-5); x≠5,-4,or-32419358699500 right 5, up 4a) 2a2b) 1681 c) 1xExplicit: f(n)=2812n-1Recursive: fn=12n-1;n≥2 and f1=28a) x=2 and-1;x≠-2, 0b) x=-5;x≠6, -3a) decay; 10% b) growth; 30% c) decay; 25%a) 24=x+1b) x-22=1625 volunteersDomain: -∞,∞, Range: -2,∞ 1943735889000048. x+1x-149. ≈ 5.8 years50. A) Exponential B) Linear C) Quadratic51. y=14.9800.938x; 6.97 mg after 12 hours52. a) not possible, the argument must be positive b) 3c) 5 ................
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