Intervals & Equations Algebra Calculus Worksheet

Here is the first partCCV / Final Exam Review – College Algebra
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the equation.
1) 7x – 11 = 0
1)
2) 9x – (4x – 1) = 2
3)
2)
2x
x
=4+
5
3
3)
4
2
=
9x 7
4)
4) 1 –
5) x(1 + 3x) = (3x – 1)(x – 3)
6)
5)
6x + 6 9x + 5
=
2x – 5 3x + 2
6)
Find the real solutions of the equation.
7) 9x – 8 = 8
7)
8)
26x – 39 = x + 5
8)
9)
2x + 3 –
9)
x+1=1
10) (x + 3)1/3 = -2
10)
Solve the inequality. Express your answer using interval notation.
11) x – 3 < 3 12) -3x - 5 -4x - 11 13) x 2 4+ 11) 12) x 10 13) 1 Solve the problem. 14) The manager of a coffee shop has one type of coffee that sells for $9 per pound and another type that sells for $13 per pound. The manager wishes to mix 50 pounds of the $13 coffee to get a mixture that will sell for $10 per pound. How many pounds of the $9 coffee should be used? 14) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 15) Susan purchased some municipal bonds yielding 7% annually and some certificates of deposit yielding 9% annually. If Susan's investment amounts to $19,000 and the annual income is $1590, how much money is invested in bonds and how much is invested in certificates of deposit? A) $13,000 in bonds; $6000 in certificates of deposit B) $5500 in bonds; $13,500 in certificates of deposit C) $13,500 in bonds; $5500 in certificates of deposit D) $6000 in bonds; $13,000 in certificates of deposit SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 16) Don James wants to invest $62,000 to earn $6320 per year. He can invest in B-rated bonds paying 13% per year or in a Certificate of Deposit (CD) paying 7% per year. How much money should be invested in each to realize exactly $6320 in interest per year? 16) 17) How much pure acid should be mixed with 9 gallons of a 50% acid solution in order to get an 80% acid solution? 17) 18) An airplane flies 490 miles with the wind and 330 against the wind in the same length of time. If the speed of the wind is 30, what is the speed of the airplane in still air? 18) 19) Gary can hike on level ground 3 miles an hour faster than he can on uphill terrain. Yesterday, he hiked 31 miles, spending 2 hours on level ground and 5 hours on uphill terrain. Find his average speed on level ground. 19) 20) An experienced bank auditor can check a bank's deposits twice as fast as a new auditor. Working together it takes the auditors 6 hours to do the job. How long would it take the experienced auditor working alone? 20) Find the slope of the line containing the two points. 21) (4, -9); (-3, 6) 21) 2 15) Find the slope of the line. 22) 22) Graph the line containing the point P and having slope m. 13 23) P = (4, 6); m = 4 23) 24) P = (0, 5); m = 1 24) 3 25) P = (-1, -10); slope undefined 25) Find an equation for the line with the given properties. 3 26) Slope undefined; containing the point - , 7 4 26) Find the slope-intercept form of the equation of the line with the given properties. 27) Slope = 0; containing the point (-6, -5) 27) Find the equation of the line in slope-intercept form. 28) 28) Solve. 29) A faucet is used to add water to a large bottle that already contained some water. After it has been filling for 3 seconds, the gauge on the bottle indicates that it contains 11 ounces of water. After it has been filling for 11 seconds, the gauge indicates the bottle contains 27 ounces of water. Let y be the amount of water in the bottle x seconds after the faucet was turned on. Write a linear equation that relates the amount of water in the bottle,y, to the time x. Find an equation for the line, in the indicated form, with the given properties. 30) Containing the points (8, 0) and (0, -12); general form 4 29) 30) Find the slope-intercept form of the equation of the line with the given properties. 31) Slope = 5; containing the point (-2, -5) Solve. 31) 32) Each day the commuter train transports x passengers to or from the city at $1.75/passenger. The daily fixed cost for running the train is $1200. Write an equation that relates the daily profit, P, in dollars to the number of passengers each day. Then use the equation to find the daily profit when the train has 920 passengers in a day. Find the general form of the equation for the line with the given properties. 3 33) Slope = - ; containing the point (5, 2) 5 Find an equation for the line with the given properties. 34) Perpendicular to the line y = 8; containing the point (2, 3) 35) Parallel to the line y = -6; containing the point (8, 2) 32) 33) 34) 35) The graph of a function is given. Decide whether it is even, odd, or neither. 36) 36) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Match the correct function to the graph. 37) A) y = -2x2 37) B) y = 1 - x2 C) y = -2x2 - 1 5 D) y = -2x2 + 1 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Determine the slope and y-intercept of the function. 38) f(x) = 12x - 6 38) Use the slope and y-intercept to graph the linear function. 39) g(x) = -3x - 2 39) 40) G(x) = -4x 40) Graph the function. State whether it is increasing, decreasing, or constant.. 41) f(x) = 5x + 3 6 41) Solve the problem. 42) You have 296 feet of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. Determine the quadratic function whose graph is given. 43) 42) 43) Vertex: (- 1, 4) y-intercept: (0, 3) Solve the problem. 44) The owner of a video store has determined that the cost C, in dollars, of operating the store is approximately given by C(x) = 2x2 - 20x + 600, where x is the number of videos rented 44) daily. Find the lowest cost to the nearest dollar. Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find that value. 45) f(x) = x2 + 1 45) Find the vertex and axis of symmetry of the graph of the function. 46) f(x) = x2 + 6x Solve the problem. 47) Regrind, Inc. regrinds used typewriter platens. The variable cost per platen is $1.40. The total cost to regrind 90 platens is $400. Find the linear cost function to regrind platens. If reground platens sell for $10.90 each, how many must be reground and sold to break even? 46) 47) 48) To convert a temperature from degrees Celsius to degrees Fahrenheit, you multiply the temperature in degrees Celsius by 1.8 and then add 32 to the result. Express F as a linear function of c. 48) 49) Suppose that f(x) = -x - 3 and g(x) = x - 12. (a) Solve f(x) = 0. (b) Solve g(x) = 0. (c) Solve f(x) = g(x). 49) 7 State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. 50) f(x) = 3x + 4x2 50) Use transformations of the graph of y = x 4 or y = x 5 to graph the function. 51) f(x) = (x - 4)4 52) f(x) = (x + 4)5 + 3 51) 52) Analyze the graph of the given function f as follows: (a) Determine the end behavior: find the power function that the graph of f resembles for large values of |x|. (b) Find the x- and y-intercepts of the graph. (c) Determine whether the graph crosses or touches the x-axis at each x-intercept. (d) Graph f using a graphing utility. (e) Use the graph to determine the local maxima and local minima, if any exist. Round turning points to two decimal places. (f) Use the information obtained in (a) - (e) to draw a complete graph of f by hand. Label all intercepts and turning points. (g) Find the domain of f. Use the graph to find the range of f. (h) Use the graph to determine where f is increasing and where f is decreasing. 53) f(x) = -x2 (x - 1)(x + 3) 53) 8 Graph the function using transformations. 3 54) f(x) = (4 + x)2 54) Graph the function. 55) f(x) = 2x (x - 2)(x + 2) 55) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph of an exponential function is given. Match the graph to one of the following functions. 56) A) f(x) = 2 x + 2 B) f(x) = 2 x + 2 C) f(x) = 2 x - 2 9 D) f(x) = 2 x 56) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use transformations to graph the function. Determine the domain, range, and horizontal asymptote of the function. 57) f(x) = 5 (x - 2) 57) Graph the function. 58) f(x) = -1 + ex 58) Solve the equation. 59) log10 x2 = 4 59) Solve the problem. 60) The long jump record, in feet, at a particular school can be modeled by f(x) = 18.9 + 2.4 ln (x + 1) where x is the number of years since records began to be kept at the school. What is the record for the long jump 21 years after record started being kept? Round your answer to the nearest tenth. Solve the equation. 61) log 5 (x - 3) = 1 60) 61) Find the exact value of the logarithmic expression. 1 62) log6 36 62) 10 Change the exponential expression to an equivalent expression involving a logarithm. 63) 5 3 = 125 Change the logarithmic expression to an equivalent expression involving an exponent. 1 = -4 64) ln e4 Solve the equation. 65) 2 1 + 2x = 32 64) 65) Find the domain of the rational function. 6x 66) f(x) = x-4 66) Find the vertical asymptotes of the rational function. -2x(x + 2) 67) f(x) = 4x2 - 3x - 7 68) h(x) = 63) 67) 4x x-9 68) Determine the maximum number of turning points of f. 69) f(x) = (x - 4)2 (x + 5)2 69) Find the x- and y-intercepts of f. 70) f(x) = (x - 5)(x - 6) 70) Find the amount that results from the investment. 71) $1,000 invested at 12% compounded annually after a period of 3 years 71) Find the effective rate of interest. 72) 8.1% compounded continuously 72) Find the present value. Round to the nearest cent. 73) To get $10,500 after3 years at 10% compounded annually 73) Solve the problem. 74) If Emery has $1600 to invest at 5% per year compounded monthly, how long will it be before he has $2300? If the compounding is continuous, how long will it be? (Round your answers to three decimal places.) Solve the problem. Round your answer to three decimals. 75) How long will it take for an investment to double in value if it earns 7.5% compounded continuously? 11 74) 75) Answer Key Testname: UNTITLED2 1) 11 7 2) 1 5 3) {60} 28 4) 45 5) 3 11 6) - 37 65 7) {8} 8) {8} 9) {3, -1} 10) {-11} 11) (- , 6) 12) [-6, ) 13) [10, ) 14) 150 lb 15) D 16) $33,000 in B-rated bonds and $29,000 in a CD 17) 13.5 gal 18) 153.75 mph 4 19) 6 mph 7 20) 9 hr 15 21) 7 22) 7 12 Answer Key Testname: UNTITLED2 23) 24) 25) 26) x = - 3 4 27) y = -5 28) y = - 2x - 5 29) y = 2x + 5 30) 12x - 8y = 96 31) y = 5x + 5 32) P = 1.75x - 1200; $410 33) 3x + 5y = 25 34) x = 2 35) y = 2 13 Answer Key Testname: UNTITLED2 36) even 37) D 38) m = 12; b = -6 39) 40) 41) increasing 42) 74 ft by 74 ft 43) f(x) = -x2 - 2x + 3 44) $550 45) minimum; 1 46) (-3, -9); x = -3 47) C(x) = 1.40x + 274; 29 platens 48) F(c) = 1.8c + 32 49) (a) x = -3; (b) x = 12; (c) x = 4.5 14 Answer Key Testname: UNTITLED2 50) Yes; degree 2 51) 52) 53) (a) For large values of |x|, the graph of f(x) will resemble the graph of y = -x4 . (b) y-intercept: (0, 0), x-intercepts: (-3, 0) , (0, 0), and (1, 0) (c) The graph of f crosses the x-axis at (1, 0) and (-3, 0) and touches the x-axis at (0, 0). (e) Local maxima at (-2.19, 12.39) and (0.69, 0.55); Local minimum at (0, 0) (f) (g) Domain of f: all real numbers; range of f: (- , 12.39] (h) f is increasing on (- , -2.19) and (0, 0.69); f is decreasing on (-2.19, 0) and (0.69, ) 15 Answer Key Testname: UNTITLED2 54) 55) 56) D 57) domain of f: (- , ); range of f:(0, ) horizontal asymptote: y = 0 16 Answer Key Testname: UNTITLED2 58) 59) {100, -100} 60) 26.3 ft 61) {8} 62) -2 63) log 5 125 = 3 1 64) e-4 = e4 65) {2} 66) {x|x 4} 7 67) x = , x = -1 4 68) x = 9 69) 3 70) x-intercepts: 5, 6; y-intercept: 30 71) $1404.93 72) 8.437% 73) $7888.81 74) 7.273 yrs, 7.258 yrs 75) 9.242 years 17