Name:. Math 0- Formula Sheet Sequences and Series t n = t + ( n )d S n = n È t ÎÍ + ( n )d S n = n Ê Á t + t n ˆ t n = t r n Ê t r n ˆ Á S n =, r r S n = rt n t r, r S = t r, r Trigonometry Exponent Laws a m a n = a m + n a m a n = a m n, a 0 Ê a m ˆ n Á = a mn ( ab) m = a m b m Ê a ˆ m Á b = a m b, b 0 m a n = a n, a 0 a a n n = a m n n = a m or Ê n Á a ˆ m sin A = sin B = sin C a b c a sin A = b sin B = c sin C c = a + b ab cos C cos C = a + b c ab Quadratic Functions y = aê Áx pˆ + q y = ax + bx + c Quadratic Equations Given ax + bx + c = 0, a 0 then Radicals Ê k ˆÊ k m a Á n b Á k m a = m k k n b n Fractions a b + c d a b c d = ac bd a b c d ˆ = mn a b = ad + bc bd = a b d c k ab x = b ± b 4ac a
Name: Multiple Choice. The common difference in the arithmetic sequence 4 5, 3 0, 9 5, 3 0, 4 5,... is 6 A. B. C. 4 D. 5. Determine the next three terms on the following sequence? 39, 7, 5, F. 3, 8, 9 G. 3, 0, H. 3, 9, J. 3, 9, 3. Find the first five terms for the following arithmetic sequence t n = 43 3n A. 30, 7, 4, 7, 30 C. 43, 30, 7, 4, 9 B. 30, 9, 8, 7, 6 D. 30, 7, 4, 9, 4. Determine the number of terms in the sequence 48, 35,,, 343 F. 3 G. 09 H. 08 J. 0 5. Complete the following arithmetic sequence. 83,,,8, A. 58; 33; 7 B. 57; 33; 43 C. 58; 33; 6 D. 57; 3; 43 6. Which term of the arithmetic sequence 6, 6, 8, has a value of 40? F. t 0 G. t 9 H. t J. t 40 7. Determine the general term of the sequence 3,, 7,, 7,... A. t n = 5n + 8 B. t n = 5n + C. t n = 5n + 8 D. t n = 5n + 8. What is the 30 th term of the sequence 3, 3.6, 4., 4.8, 5.4,? F. 00.4 G. 0.6 H. 0.6 J. 54.4 9. The sum of an arithmetic series where t =, d =, and n = 49 is A. 485 B. 4753 C. 97 D. 4753 0. Determine the sum of the arithmetic series. 5 + 3 + + ë + 07 F. 856 G. 805 H. 88.5 J. 907. The common ratio for the geometric sequence 7, 7 3, 7 9, 7 7,... is A. 3 B. 3 C. 3 D. 3 3
Name:. In the formula for the general term of a geometric sequence t n = 4 5 n Ê ˆ, the common ratio is Á 9 9 5 F. 9 G. H. J. 5 5 9 3. The sum of an infinite geometric series is 6 and its common ratio is 3. What is the first term of the series? 7 3 7 69 A. B. 46 C. D. 7 6 4. In an arithmetic sequence, t 8 = 79 and t 45 = 30. What is the value of t? F. 43 G. 49 H. 37 J. 30 5. Which of the following is a geometric sequence? A. 9, 3.5, 0.5 B. 9, 3.5, 0.5 C. 9,, 5 D. 9,, 5 6. State the common ratio of the geometric sequence. 88, 43, 648, F. G. 3 3 H. 3 J. 88 7. Complete the following geometric sequence.,,,6, A. 8; 8; 6 B. 4; 8; 3 C. 8; 8; 6 D. 4; 8; 3 8. Determine the sum (to 3 decimal places) of the first 7 terms of the geometric series: + 36 7 + 08 49 + ë F. 0.944 G. 0.944 H. 0.976 J. 0.990 9. Determine the sum of the infinite geometric series: 6 + 4 + + 4 + A. S = 5.3 B. S =.3 C. S = 4 D. S = 64 0. A geometric series has r = 3 5 and S = 3.5. Determine t. F. t = G. t = 5 H. t = 5 J. t = 3 4
Sequences and Series RETest Worksheet Answer Section SHORT ANSWER 744. 495. 8 678 OTHER. Math 0- Formulas MULTIPLE CHOICE. A. J 3. D 4. H 5. A 6. G 7. A 8. F 9. D 0. J. B. H 3. B 4. H 5. B 6. H 7. B 8. F 9. B 0. G
Trigonometry RE-test Worksheet Math 0- Name:. Given sinθ = 0.7547, where 0 θ < 360, determine the measure of θ, to the nearest degree.. Given cos θ = 0.9986, where 0 θ < 360, determine the measure of θ, to the nearest degree.. Calculate the length of AC in BAC to decimal place. (The diagram is NOT drawn to scale) 3. Given cos θ = 0.340, where 0 θ < 360, determine the measure of θ, to the nearest degree. 4. Given tanθ = 0.405, where 0 θ < 360, determine the measure of θ, to the nearest degree. 5. Given tanθ =.460, where 0 θ < 360, determine the measure of θ, to the nearest degree. 6. Given the angle 79 is in standard postions. Determine the reference angle. 3. Calculate the measure of A in CBA to the nearest tenth of a degree. (The diagram is NOT drawn to scale) 7. Given the angle 66 is in standard postions. Determine the reference angle. 8. Given that an angle has a reference angle of 54, determine the angle in standard position if the angle is in quadrant one. 9. Given that an angle has a reference angle of 68, determine the angle in standard position if the angle is in quadrant three. 0. Determine the exact value of sinθ if the terminal arm of an angle in standard position passes through the point Ê Á8, 8ˆ. 4. Calculate the length of AB in CAB to decimal place. (The diagram is NOT drawn to scale). Determine the exact value of cos θ if the terminal arm of an angle in standard position passes through the point Ê Á 9, 4ˆ.
Name: ID: A 5. Given that B is obtuse, calculate the measurement of B in BAC to decimal place. (The diagram is NOT drawn to scale) 6. Given C = 47, b = 6, c = 5 in ABC, calculate two possible measurements of A to decimal place. [ marks] 7. Given C = 8, b = 5, c = 8 in ABC, calculate two possible measurements of A to decimal place. [ marks] 8. Sarah and Simone are walking in a walk-a-thon down a straight street that leads to the finish line. At the same time, they both notice a tethered hot-air balloon directly over the finish line. Sarah sees that the angle from the ground to the balloon as 0, and Simone (who is 0.53 km closer to the finish line than Sarah) sees the angle from the ground to the balloon as 56. Determine the height of the balloon, to the nearest hundredth of a kilometre. 9. Two airplanes leave the Hay River airport in the Northwest Territories at the same time. One airplane travels at 370 km/h. The other airplane travels at 50 km/h. About h later, they are 930 km apart. Determine the angle between their paths, to the nearest degree.
Trigonometry RE-test Worksheet Answer Section. θ = 49, θ = 3. θ = 77, θ = 83 3. θ = 0, θ = 50 4. θ = 7, θ =35 5. θ = 4, θ =94 6. 8 7. 86 8. 54 9. 48 0. sinθ =. cos θ = 9 97. b = 9.4 3. A =.6 4. c = 3.5 5. angle B = 33.0 6. A = 8.7, or 4.3 7. A = 90.3, or 33.7 8. The height of the balloon is 0.6 km. 9. 60
Math 0- Name: Quadratic Functions: Retest Worksheet Short Answer. Given the equation y = x + 4x + 3, determine the following: 3. The graph of a quadratic function is shown below. Determine its equation: a) y-intercept: b) x-intercept(s): c) vertex: d) axis of symmetry: e) domain: e) range:. Given the equation y = x 6x + 8, determine the following: a) y-intercept: 4. The graph of a quadratic function is shown below. Determine its equation: b) x-intercept(s): c) vertex: d) axis of symmetry: e) domain: e) range:
Name: ID: A 5. Change the equation y = 6x 7x + to the form y = aê Áx pˆ + q by completing the square. 7. Change the equation y = 4x 6x + 9 to the form y = aê Áx pˆ + q by completing the square... 6. Change the equation y = x 4x + 6 to the form y = aê Áx pˆ + q by completing the square. 8. Change the equation y = 3x + 30x + 79 to the form y = aê Áx pˆ + q by completing the square...
Name: ID: A 9. Given the vertex is Ê Á, 3ˆ and a point on the graph is Ê Á4,4ˆ. Determine the equation of the parabola in the form y = aê Áx pˆ + q. A store sells energy bars for $.5. At this price, the store sold an average of 0 bars per month last year. The manager has been told that for every 5 decrease in price, he can expect the store to sell eight more bars monthly. a) Write a quadratic function you can use to model this situation? b) Determine the maximum revenue the manager can expect the store to achieve. c) What price will give that maximum?. 0. Given the vertex is Ê Á, 6ˆ and a point on the graph is Ê Á, ˆ. Determine the equation of the parabola in the form y = aê Áx pˆ + q. A dinner theatre has 600 season ticket holders. The owners of the theatre have decided to raise the price of a season ticket from the current price of $400. According to a recent survey of season ticket holders, for every $50 increase in the price, 30 season ticket holders will not renew their seats. a) Write a quadratic function you can use to model this situation? b) Determine the maximum revenue the concert could achieve. c) What should the owners charge for each season ticket in order to maximize their revenue?. 3
Name: ID: A 3. A hospital sells raffle tickets to raise funds for new medical equipment. Last year, 000 tickets were sold for $0 each. The fund-raising coordinator estimates that for every $ decrease in price, 00 more tickets will be sold. a) What decrease in price will maximize the revenue? 5. A rectangular dog pen is to be fenced with 8 m of fencing. Determine the maximum area and the width of this rectangle. b) What is the price of a ticket that will maximize the revenue? 6. A rectangular lot is bordered on one side by a building and the other 3 sides by 600 m of fencing. Determine the area of the largest lot possible. c) What is the maximum revenue? Show and explain your work. 4. Three rectangular areas are being enclosed along the side of a building, as shown. There is 64 m of fencing material. Assume that all the material is used. 7. A science museum wants to build an outdoor patio. The patio will be bordered on one side by a wall of the museum and the other 3 sides by 40 m of fencing. Determine the area of the largest patio possible. 8. Two numbers have a difference of 6 and their product is a minimum. Determine the numbers. a) Write the function that represents the total area in terms of the distance from the wall. 9. Two numbers have a difference of 8. The sum of their squares is a minimum. Determine the numbers. b) Determine the maximum area. 0. The sum of two numbers is. Their product is a maximum. Determine the numbers. c) Determine the length and width of the overall enclosure. 4
Quadratic Functions: Retest Worksheet Answer Section SHORT ANSWER. a) 3 b), 3 c) Ê Á, ˆ d) x = e) y. a) 8 b), 4 c) Ê Á3, ˆ d) x = 3 e) y 3. y = x 4. y = 0.5x + x 5. y = 6( x 6) 4 6. y = ( x ) + 4 7. y = 4( x ) + 3 8. y = 3( x + 5) + 4 9. y = 3( x ) 3 0. y = x + ( ) + 6. a) R = (.5 0.05x)(0 + 8x) b) $360 c) $.50 (x is 5). a) R = (400 + 50x)(600 30x) b) $94 000 c) $700 (x is 6)
3. Determine an equation to represent the situation. For each $ decrease in price, 00 more tickets will be sold. Let x represent the number of $ decreases in the price of a ticket. When the price decreases by $ x times: the price, in dollars, of a ticket is 0 x. the number of tickets sold is 000 + 00x. the revenue, in dollars, is (0 x)(000 + 00x). Let the revenue be R dollars. An equation is: R = (0 x)(000 + 00x) Use a graphing calculator. Graph: R = (0 x)(000 + 00x) Use the CALC function to determine the coordinates of the vertex. a) From the graph, the maximum revenue occurs when the number of $ decreases is 5. So, the decrease in price that will maximize the revenue is $5. b) The price of a ticket that will maximize the revenue is: $0 $5 = $5 c) Substitute x = 5 in R = (0 x)(000 + 00x) to determine the maximum revenue. R = (0 5)(000 + 00(5)) R = 45 000 The maximum revenue is $45 000. 4. a) A = 3( 64 4d)d or A = 9d d b) 768 m² c) 8 m 3 m 5. A = 49 m ; w = 7 m 6. 45 000 m 7. 00 m 8. 3 and 3 9. 4 and 4 0. 6 and 6
Math 0- Name: Absolute Value and Reciprocal Functions RE-Test workheet. Evaluate + 7 8 ( 3) + 8 + 5. 4. Write an equation for the absolute value function. Show your work... Draw the graph of y = x 3. 3. Given f(x) = x + 5x + 4, sketch a graph of the reciprocal function y = and identify the f(x) vertical asymptotes, if they exist. 5. Determine the equation of the following graph. Show your work.
Name: ID: A 6. Solve this equation: 7 8 x 3 = 9. Given the quadratic function y = f( x) below, sketch the graph of y =. Use the same coordinate f( x) plane shown.. 7. Solve this equation: x x 65 = 0. 8. Write this absolute value function in piecewise notation. y = (x + 3) 9 0. The cross-section of the sloping roof of a house is represented on a coordinate grid so that the points representing the bottom of the roof lie on the x-axis. The equation of the function describing the cross-section is h( x) = 3 x + 4, where h is the height of the roof, in metres, and x is the horizontal distance from the centre of the roof, in metres. What is the width of the bottom of the roof?
Absolute Value and Reciprocal Functions RE-Test workheet Answer Section. 55. 3. The graph of y = f(x) opens up and has x-intercepts and 4. So, the graph of the reciprocal function has vertical asymptotes x = and x = 4. Plot points where the lines y = and y = intersect the graph of y = f(x). These points are common to both graphs. Using these points and the asymptotes, draw smooth curves that approach the asymptotes but never touch them.the graph of the reciprocal function has Shape 3.
4. Choose two points on the line to determine the slope of the linear function: ( 4, 5) and (, ) m = y y x x m = (5) ( 4) m = y-intercept: 3 An equation for the absolute value function is: y = x 3 5. y = x 8 6. x = 4 and x = 4 7 7. The solutions are: x = 5, x = 7, x = 5, and x = 3 Ï 8. y = Ô (x + 3) 9, if x 6 or x 0 Ì ÓÔ (x + 3) + 9, if 6 < x < 0 9. 0. Therefore, the distance between the two points is m. The width of the bottom of the roof is m.
Radicals Review. Write this entire radical as a mixed radical:. Write this entire radical as a mixed radical: 3. Write this entire radical as a mixed radical: 4. For which values of the variable, x, is this radical defined? 5. Simplify by adding or subtracting like terms: 6. Simplify by adding or subtracting like terms: 7. Expand and simplify this expression: 8. Expand and simplify this expression: 9. Expand and simplify this expression: 0. Expand and simplify this expression:. Simplify this expression:. Solve this equation: 3. Solve this equation: 4. Determine the root of each equation. a) b) c) d)
Answer Section (Radicals) SHORT ANSWER. ANS:. ANS: 3. ANS: 4. ANS: 5. ANS: 6. ANS: 7. ANS: 8. ANS: 9. ANS: 0. ANS:. ANS:. ANS: 3. ANS: 4. ANS: a) b) c) The equation has no real root. d)
Math 0- Name: Class: Quadratic Equations RE-Test Worksheet. Factor this polynomial expression: 6( x 4) 9Ê Á4y + 3ˆ [ marks].. Factor this polynomial expression: 4( x + 4) + 7( x + 4) + 4 [ marks]. 3. Determine the discriminant of x + 8x + 8 = 0 [ mark]
Name: ID: A 4. Describe the nature of the roots of x x + = 0 (Do not solve) [ mark] 5. Solve x + x 3 = 0 to the nearest hundredth. [ marks] 6. Determine the exact solution(s) for (x 5) = 6 [ marks] 7. Solve 6x 6x = 0 by factoring. [ marks] 8. Solve x + 7x 0 = 0 by factoring. [ marks] 9. Algebraically solve 4x + x + 3 = 0 [ marks]
Name: ID: A 0. Determine the exact roots of x x 9 = 0 [ marks]. Determine the exact roots of 9x + x 9 = 0 [ marks] Problem. A penny is dropped from the top of the High Level Bridge. It s height, h meters, above the river t seconds after it is released is modeled by the quadratic function: h( t) = 0 4.8t. To the nearest tenth of a second, how long has the penny fallen for when it is 9 m above the river? [ mark].. Marc s rectangular garden measures 9 m by m. He wants to double the area of his garden by adding equal lengths to both dimensions. Determine this length to the nearest tenth of a metre. Show your work. [ marks] 3
Quadratic Equations RE-Test Worksheet Answer Section SHORT ANSWER. Ê Á 4x + y 7 ˆ Ê Á 4x y 5 ˆ. 4(x + 8)(x + 8) 3. 0 4. Discriminant = 0 and therefore equal real roots 5. x =Ö 0.54 or 0.46 6. x = 5 ± 6 7. x = 0 or 8. x = 5 or 9. x = or 3 4 0. x = ± 433 4. x = ± 5 3 8 PROBLEM. 4. s. The length to be added is approximately 4.3 m.
Rational Expressions Multiple Choice. Which of the following are NOT rational expressions? i) ii) iii) iv) A. iii and iv B. i, iii, and iv C. ii and iii D. ii and iv. Which of the following rational expressions are defined for all real values of x? i) ii) iii) iv) A. i and ii B. iv C. i and iii D. i and iv 3. Which of the following are the non-permissible values for this rational expression? A.,, and C. and B. and D.,, and 4. Simplify. A. C. B. D. 5. Simplify. A. C. B. D.
6. Simplify. A. C. B. D. 7. Simplify this expression: A. B. C. D. 8. Solve. A. B. C. D. 9. Solve. 0. Simplify. A. C. B. D. no solution A., C., B., D., Short Answer. Simplify this expression:. Simplify.
3. A plane travels from A to B and back, a distance of about 95 each way. On the journey out, the plane has a 5 km/h tailwind, and on the return trip, it has a 0 km/h headwind. Write a simplified rational expression for the total flying time in terms of the plane s average speed in still air,. 4. Simplify. 5. Solve. 6. Simplify this expression: 7. Write the least common multiple of the expressions in each pair. a) b) c) 8. Simplify this expression and state the non-permissible values. Show your work. Rational Expressions Answer Section MULTIPLE CHOICE. B. D 3. D 4. A 5. B 6. B 7. D 8. D 9. C 0. C
SHORT ANSWER.,. 3. 4. 5. 6., 5, 7. a) b) c) 5, 8. So,
Math 0- Name: Quadratic Systems of Equations RE-Test worksheet. ( point) Solve by graphing: y = 3 4 ( x 4) 3x + y 8 = 0 3. ( points) Algebraically solve: y =8x 9 4x 4x y +35 = 0. ( point) Solve by graphing: x + 6x + 4y 7 = 0 x + y = 0.
4. ( points) Algebraically solve:: x + 8x y + = 0 3x + y + 6 = 0 5. ( point) Algebraically solve: A line with slope = and y-intercept of + intersects a parabola with vertex (, ) and a point Ê Á 3, 6 ˆ..
Problems: (Show all work). Graphical solutions require equations, sketch, and window settings.. (6 points) After a football is kicked, it reaches a maximum height of 3 m and it hits the ground 6 m from where it was kicked. After a soccer ball is kicked, it reaches a maximum height of 9 m and it hits the ground 40 m from where it was kicked. The paths of both balls are parabolas. a) Consider both balls to be kicked from the same starting point and place this at the origin of a coordinate grid. Draw a diagram to model the given information for the two kicked balls. ( point) b) Determine a quadratic equation in the form of: y = a(x p) + q to model the football height compared to the horizontal distance it travelled and a quadratic equation in the form of: y = a(x p) + q to model the soccer ball height compared to the horizontal distance it travelled. ( point) c) Solve the system of two equations. If you are using a graphical approach, make sure you include all relevant information - including the window settings and equations you used to solve the problem for full marks. Round your answer to the nearest centimeter ( points) d) What is occuring at the intersection point, in the context of this problem? ( point) 3
. (5 points) The perimeter of the right triangle is 50 m. The area of the triangle is 30y square metres. a) Write a simplified expression for the triangle s perimeter in terms of x and y. ( point) b) Write a simplified expression for the triangle s area in terms of x and y. ( point) c) Write a system of equations and explain how it relates to this problem. ( point) d) Solve the system for x and y. What are the dimensions of the triangle? ( points) 4
Other. (0 points) Math 0- Formula Sheet Sequences and Series t n = t + ( n )d S n = n È t ÎÍ + ( n )d S n = n Ê Á t + t n ˆ t n = t r n Ê t r n ˆ Á S n =, r r S n = rt n t r, r S = t r, r Trigonometry Exponent Laws a m a n = a m + n a m a n = a m n, a 0 Ê a m ˆ n Á = a mn ( ab) m = a m b m Ê a ˆ m Á b = a m b, b 0 m a n = a n, a 0 a a n n = a m n n = a m or Ê n Á a ˆ m sin A = sin B = sin C a b c a sin A = b sin B = c sin C c = a + b ab cos C cos C = a + b c ab Quadratic Functions y = aê Áx pˆ + q y = ax + bx + c Quadratic Equations Given ax + bx + c = 0, a 0 then Radicals Ê k ˆÊ k m a Á n b Á k m a = m k k n b n Fractions ˆ = mn a b a b + c ad + bc = d bd a b c d = ac bd a b c d = a b d c k ab x = b ± b 4ac a 5
Quadratic Systems of Equations RE-Test worksheet Answer Section SHORT ANSWER. P(,) P(4, ). P(3, 5) P(,3) 3. Solution: Ê Á4,3ˆ 4. P( 7,5) P( 4, 4) 5. y = x+ y = ( x ) Solution: Ê Á, 3ˆ
PROBLEM. a. b. football: y = 5 x( x 6) soccerball: y = 9 600 x( x 40) c. (0., 9.00) d. The intersection point is where the two balls are at the same height given the same horizontal distance travelled.. a) 8x + y + 9 b) 5 x + 363 x c) y = 8x + 38 y = 4 x + 0 x d) x = 0 and y = base =, height = 60, hypotenuse = 9 OTHER. Math 0- Formulas