MPM2D Trigonometry Review

1. What are the three primary trig ratios for each angle in the given right triangle?

2. What is cosθ?

3. For the following triangles, if ΔABC~ΔDFE, state a)the ratio of side lengths in lowest terms and b) the ratio of areas.

        4. For the following triangles, find the indicated measurement. You may need to use Sine, Cosine or Tangent

ratios, Sine Law, Cosine Law or properties of similar triangles. a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

5. a) draw and label a triangle that you can solve for a missing angle using Sine Law. b) draw and label a triangle that you can solve for a missing side using Sine Law. c) draw and label a triangle that you can solve for a missing angle using Cosine Law. d) draw and label a triangle that you can solve for a missing side using Cosine Law. 6. Sarah and Rachel stood at opposite ends of Mitchell’s main street, about 950 m apart. A hot air balloon hovered

over main street between them. Rachel measured an angle of elevation of 720 to the balloon while Sarah measured a 590 angle of elevation. How high above main street was the balloon?

7. A boat dropped its anchor and let it out it’s full 450m of chain. It then floated until the chain is taut and

measures a 700 angle of depression from where it leaves the boat. If it leaves the boat 3 meters about water level, how deep is the water?

8. A post is supported by two wires, one on each side. The wires are anchored to the ground 12 m apart so that

they create 400 and 600 angles with the ground. How long are each of the wires? 9. Two ships sail from Halifax. The Nina sails due east for an hour for a total of 115 km. The Pinta set sail a the

same time heading 430 south of east, and after an hour had traveled 98 km. How far apart are the ships? 10. Two scuba divers are 20 m apart when they spot a shark swimming along the bottom. They measure that angle

of depression from each of them to be 470 and 400. a) How far is the straight line distance from each diver to the shark? b) If the divers are at a depth of 20 m, how deep is the water?

Analytic Geometry

1. State an equation of a line that is parallel to

€

y = 5x + 4 .

2. State an equation of a line that is perpendicular to

€

y =14x + 3.

3. What is the equation of a vertical line through (9, 10)? 4. Graph the following lines

a)

€

y = −3x + 6 b)

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y =14x − 2 c)

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y = x d)

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y = 5 e)

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y = 4 − 23x

f)

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x + y = 7 g)

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2x + 6y −12 = 0 h)

€

x = −2 5. State the equations of the following lines, first in slope-intercept then in standard form.

6. Find the standard form equation of a line that passes through (-2,7) with a slope of

€

−38

.

7. Find the equation in standard form of a line that passes through (2,7) and (-6, 11). 8. Find the equation in standard form of a line parallel to

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2x + 3y − 7 = 0 passing through (6,7). 9. Find the equation of the line passing through (-2, -5) that is perpendicular to

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4x + 6y + 2 = 0 10. A line segment has endpoints (-3, 4) and (6, 22) a) calculate its slope b) calculate its midpoint c) calculate its length d) write its equation in standard form 11. A triangle has verticies (-5, -6) (-2, 6) and (10, 3) a) Classify the triangle as equilateral, isosceles or scalene. b) Determine if the triangle has a right angle. 12. If a line segment AB has a midpoint (4, 2) and A is (9, 12). What is the midpoint?

13. Find endpoints of a line segment so that it has a slope of

€

25

.

14. An airport tower is located in Toronto. It locates two airplanes approaching. One plane is 85 km east and 95 km north of Toronto. The other is 64 km west and 115 km north of Toronto.

a) How far apart at the two planes? b) If the two planes are travelling at the same speed, which one touches down at the airport first?

Systems of Equations 1. State the equations of a pair of parallel lines. 2. State the equations of a pair of perpendicular lines. 3. What is the solution of the given system

4. How many solutions does a) a dependent system have; b) a consistent system have? 5. Sketch an inconsistent system. 6. Determine the number of solutions for the following system. Justify your conclusion.

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2x + 3y = 9

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y = 3− 23x

7. Solve the following systems by substitution or elimination.

a)

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2x + y = 33x + 2y = 5

b)

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3x = 2y −10y +15= 3x

c)

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x − 3y = 112x = −10y + 6

d)

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6x = 12− 3yy − 2x = −16

e)

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2x − 3y = 135x − y = 13

f)

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3x + 21= 5y4y + 6 = −9x

g)

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8x − 3y = 226x +12y = −12

h)

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x2

+y8

= 4

x3−y2

= −2

8. Students from the local high school are selling tickets to the town's annual carnival. Adult admission is $5.00

and child admission is $2.50. Two hours after the carnival opened its first day, 440 tickets had been sold totaling $1900. How many adults and how many children entered the carnival during the first two hours it was open

9. Flying to Kampala with a tailwind a plane averaged 158 km/h. On the return trip the plane only averaged 112

km/h while flying back into the same wind. Find the speed of the wind and the speed of the plane in still air. 10. A boat traveled 210 miles downstream and back. The trip downstream took 7 hours. The trip back took 10

hours. What is the speed of the boat in still water? What is the speed of the current? 11. At Elisa's Printing Company LLC there are two kinds of printing presses: Model A which can print 70 books per

day and Model B which can print 55 books per day. The company owns 14 total printing presses and this allows them to print 905 books per day. How many of each type of press do they have?

12. Cream has a fat content of 15% and milk has a fat content of 3%. How much milk and cream need to be mixed

to create 20 L of cream that is 6%? 13. Create a graph for each type of linear system (dependant, consistent and inconsistent) and determine the

equations of the lines you used for each.

Polynomials 1. What is the degree of the polynomial

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2x(x2 + 3)(x − 4)(x + 6) 2. Give an example of a trinomial that has a degree of 6. 3. Expand and simplify each of the following.

a)

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(2x + 3)(5x − 6) b)

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(4x + 5)2 c)

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2x(x − 4)(2x + 7) d)

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(2x + 5)(2x − 5)

e)

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3+ 2x(x − 4) − (3x − 2)2 f)

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3(2x + 4)(x + 3) − 4x(3x − 4)2 4. Factor the following fully.

a)

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80v2u − 8v3 + 40v2 b)

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−4 x5y + 20x4 yz − 24 x3y3 + 28x3yz c)

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−32mn8 + 4m6n +12mn4 +16mn 5. Factor the following completely.

a)

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x2 + 4x − 32 b)

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21xy +15x + 35ry + 25r c)

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3x2 − 3x − 90 d)

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−2x3 − 6x2 + 56x

e)

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6x2 −11x + 4 f)

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15x2 +14x − 8 g)

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8x3 − 64x2 + x − 8 h)

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5x2 − 50x +120

i)

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9x2 +12x + 4 j)

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24x2 − 6xy − 9y2 k)

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24x2y + 34xy +12y l)

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12x2 + 5x − 2

m)

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21x3 − 84x2 +15x − 60 n)

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2x2 −18 n)

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16x4 − 81y8 o)

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x4 − y4

p)

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m2 − (2+ 3m)2 q)

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9x2 − 4y2 r)

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a(3x − y) + b(3x − y) − 5(3x − y)

6. Explain how you know a SUM of SQUARES is not factorable, but a DIFFERENCE of SQUARES is. 7. A rectangle has a length that is 5 greater than 3 times it’s width. a) Sketch and label the rectangle b) Find and simplify an expression for the area of the rectangle. c) Find and simplify an expression for the perimeter of the rectangle. 8. Find and simplify and algebraic expression for the area of the following figure. Use your SIMPLIFIED

expression to determine the area when the radius of the semi circle is 3 ft.

9. a)The trinomial

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x2 + 6x + 4 is NOT a perfect square. Change ONE term to turn it into a perfect square. b) The trinomial

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6r2 − 4r +1 is NOT a perfect square. Change ONE term to turn it into a perfect square. 10. Find, simplify and factor an expression for the UNSHADED region of the figure below.

11. The AREA of a rectangular field can be represented by the expression

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6x2 + 7x − 20 . a) Factor this expression to determine an expression for each of length and width. How can you tell which is

length and which is width? Explain your answer fully. b) If x must be a positive whole number, what is the smallest value it can be? How did you determine this

answer? c) If x = 20 m, calculate the PERIMETER of the field.

Quadratic Functions 1. State the domain and range of each of the given relations.

a) b) c) d)

D: { } D: { } D: { } D: { } R: { } R: { } R: { } R: { }

e) { (0,4) (1,3) (6,8) (7,11) (2,8) } D: { } R: { } f) { (4,3) (7,8) (3,-2) (4,6) (7,8) } D: { } R: { } 2. State the equation of each parabola a) has a vertex of (3, -6) ____________________________

b) congruent to

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y =12x2 with x=7 as the axis of symmetry and a maximum value of 4 ____________________

c) stretched by 3 and shifted up 5 ____________________________ 3. Find the value of ‘k’ if

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y = 4x2 + k passes through (-1,-1) 4. Complete the following table for the given parabolas. Graph each parabola.

Vertex Axis of Symmetry Max/Min Value Range

a)

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y =12x2 + 2

b)

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y = −2(x − 3)2

c)

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y = (x + 4)2 − 6

d)

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y = −14(x +1)2 + 3

e)

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y = −3x2 +10

5. Complete the square for each of the following, then state the vertex, axis of symmetry, max/min value and

range for each.

a)

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y = 0.5x2 + 6x − 3 b)

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y = x2 +10x + 7 c)

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y = −4 x2 − 24x −12 d)

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y = −13x2 +10x + 5

6. Find the x and y-intercepts of the following parabolas.

a)

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y = 2(x + 3)2 − 32 b)

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y = 3(x − 5)2 + 7 c)

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y = −5(x +1)2 + 2 d)

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y = −12x2 +10

7. Find the x and y-intercepts of

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y = 5x2 +10x − 3 8. Explain how finite differences in a table of values can tell you whether a relations is linear, quadratic or neither. 9. Use finite differences to determine if the following tables represent linear or quadratic relations (or neither).

x y x y 0 7 2 -5 1 9 4 -11 2 15 6 -17 3 25 8 -23

10. The path of a thrown ball can be represented by

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h = −0.004d 2 + 0.112d +1.6 where h is the height of the ball in metres, and d is the horizontal distance, in metres, from the person who threw the ball.

a) How HIGH was the ball when the thrower first releases it? b) What is the MAXIMUM height of the ball? c) What is the HORIZONTAL DISTANCE the ball has travelled when it reaches its maximum height? 11. Tickets for a show at the children’s theatre cost $5 and the 120 seats in the theatre are filled daily. Added

expenses mean that the theatre owner has to increase the cost of tickets, but a survey shows that for each $0.50 increase in ticket cost, they will sell 10 fewer tickets to each show. Based on these statistics, what ticket price do you recommend in order to maximize their revenue?

12. An apple orchard now has 80 trees and each tree on average produces 400 apples. For each additional tree

planted the average number of apples per tree drops by 4. How many additional trees need to be planted to maximize the number of apples (yield) produced each year in the orchard?

13. Creasyn and Morgan are knitting scarves to sell at the craft show. The wool for each scarf costs $6. They were

planning to sell the scarves for $10 each, the same as last year when they sold 40 scarves. However, they know that if they raise the price, they will be able to make more profit, even if they end up selling fewer scarves. They have been told that for every 50¢ increase in the price, they can expect to sell four fewer scarves. What selling price will maximize their profit and what will the profit be?

14. An electronics store sells an average of 60 entertainment systems per month at an average of $800 more than

the cost price. For every $20 increase in the selling price, the store sells one fewer system. What amount over the cost price will maximize profit?

15. Arnold has 24 m of fencing to surround a garden, bounded on one side by the wall of his house. What are the

dimensions of the largest rectangular garden that he can enclose? 16. Jamie throws a ball that will move through the air in a parabolic path due to gravity. The height, h, in metres, of

the ball above the ground after t seconds can be modelled by the function

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h = 4.9t 2 + 40t +1.5 a) Find the zeros of the function and interpret their meaning. b) Determine the time needed for the ball to reach its maximum height. c) What is the maximum height of the ball?

Quadratic Equations 1. We know that the roots of a quadratic equation are the same as the x-intercepts (zeros) of the function that we

get if we replace the zero in a quadratic equation with a y. Describe the kinds of roots of quadratic equations using the cooresponding functions as references.

2. Complete the square for

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−2x 2 + 8x + c = 0 and use what you know about the cooresponding function to determine the value of c that will result in a) Equal Roots, b) Complex Roots (no real roots) and c) Two Real Roots.

3. State the solutions to the following quadratic equations that are ALREADY in factored form.

a)

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(x + 2)(x + 5) = 0 b)

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(x − 6)(2x + 3) = 0 c)

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(3x + 4)(5x − 7) = 0 d)

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2x(3x + 9) = 0 4. Solve the following quadratic equations by factoring.

a)

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4x 2 −16x = 0 b)

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x 2 + 5x + 4 = 0 c)

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3x 2 + 36x + 49 = 8x d)

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6x 2

5−2x4

= 0

e)

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3m2 = −16m − 21 f)

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3x 2 +14x − 49 = 0 g)

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3x 2 − 4 = −8x −1 h)

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2x 2 − 50 = 0

i)

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x 2 − 7x2

+32

= 0 j)

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x 2

2−5x4

= 3 k)

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5x 2 = −2x l)

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x 2 +11= 300

5. Solve using the quadratic formula. Round your answers to the nearest hundredth if necessary, otherwise

leave as a fraction.

a)

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5x 2 − 8x − 4 = 0 b)

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4x 2 − 20x + 25= 0 c)

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x 2 +14x + 9 = 0 d)

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19− 4x 2 = 2

e)

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(2x + 3)2 − 6 = 11x + 8 f)

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2x(x − 6) = 3x + 9

6. The path of a thrown ball can be represented by the relation

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h = −0.007d 2 + 0.4d +1.5 , where d is the horizontal distance traveled in metres and h is the height in metres. What is the horizontal distance the ball travels before it hits the ground?

7. A walkway around a 6 m by 9 m lawn is the same width on all 4 sides. If the area of the walkway is the same as

the area of the lawn, how wide is the walkway? 8. A football is punted, and its path is modeled by the function

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h = −0.1d 2 + 3.4d + 8 , where h metres is the height of the football and d metres is the horizontal distance from the line of scrimmage. Find, to the nearest metre, how far the punt travels from the line until it first hits the ground.

9. The area of a rectangle is 560 square inches. The length is 3 more than twice the width. Find the length and the

width. 10. In a right angle triangle the base is 3 cm less than the height and the hypotenuse is 4 cm. What are the base

and height? 11. The area of a triangle is 32 cm2. If the base is 2 cm shorter than triple the height, what are the dimensions of

the triangle?

12. A sticker warehouse sells an average of 6 rolls of stickers per customer at $4 per roll. Statistics show that for every $0.25 decrease in price, customers will buy an additional roll.

(a) According to this model, if the stickers were reduced to $3 per roll, what will be the revenue? (b) According to this model, at what sticker price will the revenue from stickers be $28.    

13. A sporting goods store sells 90 ski jackets in a season for $200 each. Each $10 decrease in the price would

result in five more jackets being sold. Find the number of jackets sold and the selling price to give revenues of $17 600 from sales of ski jackets.

14. Eduardo works in the “remodeling” division at a modern art museum. One of his jobs is to keep the different

metal sculptures painted and in good condition. One outdoor sculpture has the shape of a large right triangle. Eduardo needs to know the area of the triangle so that he can figure out how much paint it will need. He finds out that the area he needs to paint is 10 square feet. If the longer leg is 3 feet less than twice the shorter leg, what are the lengths of the two legs?