1. Graph each of the given equations, state the domain and range, and specify all intercepts and symmetry. a) y 3x

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1 MATH 94 Final Exam Review. Graph each of the given equations, state the domain and range, and specify all intercepts and symmetry. a) y x b) y x 4 c) y x 4. Determine whether or not each of the following relations represents a function and give the domain and range of each. a) {(6, ), (, -7), (, ), (4, -)} b) x y 6 c) y 4x. Evaluate each function as indicated. a) Given f i) f () ii) f (0) x 4, if x 5, if ( x 0, find: x 0 b) Given the graph of f ( below, find: i) f (4) ii) f (0) iii) f () iii) f (4) f ( x h) f ( 4. Find and simplify the difference quotient for each function below. Show work! h a) f ( x 7x b) f ( 5x c) f ( x 0 5. Determine whether each of the following functions is even, odd, or neither. a) f ( 6x 4x b) f ( 5x 6x c) y 5 x 6. Find the equation of the line in slope-intercept form (if possible) which satisfies the given conditions. a) passing through points (7, -) and (5, ) b) parallel to x 5y 0 through (5, 6) c) perpendicular to the x-axis through (, -7) 7. State the slope and y-intercept of each line whose equation is given. Then graph each line. a) 4x 5y 5 b) y 4 8. Describe how the graph of each transformation below would differ from the graph of f (. a) f ( x ) b) f c) f ( 6 ( 4

2 9. Given the graph of f (, graph each of the given transformations. a) f ( x ) b) f ( c) f ( 0. Graph the given piecewise function. f x, if x, if ( x 0 x 0. Given f ( x and g ( 5x 0, find each of the following in simplest form. a) ( f g)( b) ( f g)( c) ( f g)( f d) ( g e) ( f g)( f) ( g f )(. Given f ( x 4 and g ( x 5, find each of the following in simplest form and state their domains. a) ( f g)( b) ( g f )(. Using the functions from #, find each of the following. a) ( f g)(4) b) ( g f )(4) c) ( f g)(0) d) ( g f )(0) f e) ( f g)(4) f) ( f g)() g) ( f g)( ) h) (5) g 4. Tell whether or not each function below is one-to-one. a) y 4x 7 b) y x 5. Find the inverse of each function and state the domain and range of both the original and the inverse. Show your work! x 5 x a) f ( x 5 b) f ( c) f ( 5e x 8

3 6. Given the points (, 6) and ( 5, 8), find the (a) distance between them, and (b) midpoint of the line segment joining them. 7. Give the standard form of the equation of each of the circles, based on the information given. a) center: (4, ); radius 5 b) center: (0, 0); passing through ( 5, 7) 8. A diameter of a circle has endpoints at ( 4, ) and (0, 9). Determine a) the center of the circle, b) the radius of the circle, and c) give the equation of the circle. 9. State the center and radius of the circle whose equation is ( x 6) y 4 and graph it. Then use the graph to state the domain and range of the circle. 0. Write the given equation in standard form by completing the square, and then state the center and the radius. x y x 8y 0. A real estate agency offers its new employees a base salary of $4,000 plus a $00 bonus for every new client they sign. a) Express the annual salary, f, as a function of the number of new clients signed, x. b) How many new clients must be signed to earn a salary of $0,000?. Perform the indicated operations and express your answer in standard form. ( a bi) a) ( 7i) ( i) b) ( 6 5) ( 5 49) c) ( 9 4i)( i) d) 8 i 4 4i e) ( 6) f) Solve each quadratic equation by either factoring, using the square root property, or using the quadratic formula. You must show your work in order to get credit. Give your answers in exact simplified form, not as rounded decimals. a) x 6x 0 b) x 8x 0 c) ( x 4) 5 4. Determine the vertex, x-intercept(s) and y-intercept of the quadratic function whose equation is given. Then graph it. f ( ( x ) 8

4 5. Use the leading coefficient test to determine the end behavior of the graph of each polynomial function below a) f ( x 74x 90x 700 b) f ( 6x x 4x Determine the x-intercept(s) and y-intercept of each polynomial function. Then use these points and the end behavior to sketch a graph of the function. Be sure to notice what the behavior of the graph at each x-intercept should be. a) f ( ( x ) (x 5)( x 4) b) f ( x 6x 9x 4 7. Determine a formula for the polynomial function whose graph is shown. Assume that the scale on both axes is, and that a complete graph is given. You may leave your answer in factored form. 8. Use polynomial division to find the quotient Q ( and the remainder R ( when the following division is performed. 4 x 7 x x x 7x 9 9. Use synthetic division to show that 4 is a solution to the equation x 7x 9x 4 0, and then use the quadratic formula to find the other solutions in their exact, not decimal, form. Show your work! 0. Use the rational zero theorem to list all possible rational zeros of f (. Then determine which ones are actually zeros of f ( x 9x 8x Find all the zeros of the function f ( x x 5x 4x. State your zeros in their exact (not decimal) form and show work!. Find a polynomial of degree with real coefficients which has zeros of i and 6, given that f ( 0) 7. Write your answer in expanded, not factored, form.

5 . State all asymptotes and intercepts of the given rational functions. Then graph each function. x 6x x 4 x 9x a) y b) y c) y x 4 x 5x 4 x 4 4. Solve each inequality and state your solution in interval notation. State your boundary points and show your sign chart. x x a) x ( x 5)( x ) 0 b) 0 6 x 5. Graph the function x y and state its domain, range, and asymptote. 6. Describe how the graph of each function below would differ from the graph of a) x y 4 b) y x x y. 7. Find the accumulated value (rounded to the nearest cent) of an investment of $8000 for 40 years at an interest rate of 4.6% if the balance is compounded: a) annually b) monthly c) continuously 8. On their wedding day, a couple is given a check which they deposit into an account paying % interest compounded continuously. On their 50 th anniversary, their balance has accumulated to $5,000. How much was the original check (to the nearest cent)? 9. Find the time required (to the nearest year) for an investment of $8000 to double if it is invested at % interest, compounded annually. 40. Evaluate each logarithmic expression exactly. a) log 7 7 b) ln e c) log 5 4. Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions. 4 a) log( 00xy ) b) e x ln y 4. Use properties of logarithms to condense each logarithmic expression. Write each expression as a single logarithm whose coefficient is. a) log x log z b) log x log y 4

6 4. Solve each exponential equation algebraically. Show your work, and give both an exact answer and a decimal approximation, rounded to the nearest thousandth. a) x x x 0 b) 6(.4) 5 c) 5e Solve each logarithmic equation algebraically. Show your work! a) log 8 (x 7) b) log( 4x ) log( x ) c) log ( x ) log ( d) log 7(4x ) 5 log The function P( 95 0log x models the percentage of students who could recall the important features of a classroom lecture as a function of time, where x represents the number of days that have elapsed since the lecture was given. a) What percentage (to the nearest whole percent) of the students recall important features of the lecture after 5 days? b) When (to the nearest tenth) do only 0% of the students recall the important features of the lecture? 46. The population of Timbuktu in 940 was 5000, and in 00 it was estimated to be 5,000. kt a) Determine the growth rate, k, to 4 decimal places, and write a function of the form A A0e to model the population of Timbuktu t years after 940. b) Use your function to predict the population of Timbuktu in the year A substance decreases from 500 grams to 00 grams in 4 days. a) Determine the decay rate, k, to 4 decimal places, and write a function of the form model the quantity of the substance at time t. A e A 0 kt to b) Use your function to determine half-life of the substance, to the nearest tenth. 48. Find a positive angle less than 60 or which is coterminal with the given angle. a) 0º b)

7 49. Show the approximate location of the following points on the unit circle below. Label your points a, b, c, and d. a) P 4 b) P 7 c) P 4 7 d) P The minute hand of a clock moves from :00 to :00. a) Through how many degrees does it move? b) Through how many radians does it move? c) If the hand is inches long, what distance does the tip of the hand travel? Give the answer in terms of, and include units! 5. Complete the following table. Give exact values, not decimal approximations. t 0 cos t sin t State the exact value for each of the following. (No decimals.) a) sin b) cos c) 4 6 tan d) sec 4 5. Suppose a point P(t) on the unit circle has coordinates trigonometric function values for t. 5, 6 6. Determine all six 54. Determine the quadrant in which lies, given the following information. a) cos 0, sin 0 b) csc 0, tan 0

8 55. Use the Pythagorean Theorem to determine the exact value of the missing side of each triangle, and then determine the six trig function values for in their exact forms. Rationalize denominators when necessary. a) b) 56. Determine the amplitude and the period of each function, and then sketch one complete cycle of the graph. Be sure to show your scale on each axis! a) y sin(4 b) y 6 cos x c) y sin(x ) 57. Describe in words how the graph of each of the given functions would differ from the graph of y sin x. a) y sin( 5 b) y 9sin( x ) 58. Determine the exact value of each expression. Keep in mind that the range of y sin x is, and the range of y cos x is [0, ]. a) cos b) sin c) cos (0) 59. Construct a right triangle to find the exact value of : tan cos Construct a right triangle to find the exact value of: sec tan 6. Convert each angle from degree measure to radian measure or vice-versa. All radian measures should include. 5 7 a) 0º b) 45º c) d) 0 6. Find the length of an arc with angle 0º in a circle of radius inches, rounded to two decimal places. Give units!

9 6. For each of the right triangles below, find the lengths of the missing sides to the nearest tenth, and find the measures of the missing angles to the nearest degree. a) b) c) 64. At a certain time of day, the angle of elevation of the sun is 70. To the nearest foot, find the height of a tree whose shadow is feet long. 65. For each of the given graphs, give either a cosine function or a sine function. Note that the x-axis goes from to (scale ), and the y-axis goes from -6 to 6 (scale.) a) b) c) 66. Use the Law of Sines and/or the Law of Cosines to find the values of all sides and angles not given in the triangles below. Round all side lengths to the nearest tenth, and round all angles to the nearest degree. a) b) c) 67. Smithville is 850 miles from Davisboro, which is 95 miles from Greentown. If Smithville is 000 miles from Greentown, and the locations of the three towns form an oblique triangle, determine the angle formed at Smithville, rounded to the nearest degree.

10 68. A telephone pole is being supported by two guy wires that are attached to the top of the pole and anchored into the ground on opposite sides of the pole at points A and B, which are 60 feet apart. If the angles of elevation at A and B are 76º and 6º, respectively, find the lengths of both wires and the height of the pole, rounded to the nearest tenth. State the vertex, focus, and directrix of the parabola whose equation is given. Then sketch its graph. 69. y ( x ) x ( y 4) 4 7. x 4y 8y 0 7. y x x 7 Determine the equation of the parabola that satisfies the given conditions. 7. vertex: (0, 0); focus: (0, 4) 74. focus: (, ); directrix: y 75. vertex: (5, ); directrix: x = 76. focus: (, -); directrix: x State the center, vertices, and foci of the ellipse whose equation is given. Then sketch its graph. 77. x y ( x 5) ( y ) x y x x 4y 90x 6y 97 0

11 Determine the equation of the ellipse that satisfies the given conditions. 8. x-intercepts ( 5, 0); y-intercepts (0, 6) 8. vertices ( 4, 0); foci (, 0) 8. y-intercepts (0, 7) ; foci (, 0) 84. foci: (5, 0) & (, 0); one vertex at (0, 0) 85. Vertices at (5,4) and ( 5, ) ; length of the minor axis 4. State the center, vertices, foci, and asymptotes of the hyperbola whose equation is given. Then sketch its graph. 86. x y ( y 4) ( x ) y 4x 50y 4x x 90x 4y 8y 85 0 Determine the equation of the hyperbola that satisfies the given conditions. 90. foci at (,0) ; vertices at,0 9. foci at 0, 6; vertices at 0, 4 9. foci at (, 5) and (, -); one vertex at (, 4)