Polynomials. Lesson 6

Transcription

1 Polynomials Lesson 6

2 MPMD Principles of Mathematics Unit Lesson 6 Lesson Six Concepts Introduction to polynomials Like terms Addition and subtraction of polynomials Distributive law Multiplication and division of polynomials Simplifying expressions then using substitution Polynomials Polynomials are a mathematical expression with one or more terms, in which the exponents are whole numbers and the coefficients are real numbers. Monomial is a polynomial with one term; e.g. 8 and x. Binomial is a polynomial with two terms; e.g. 4x +7 and y. Trinomial is a polynomial with three terms; e.g. 4x x 9 + and w + 7w +. Term -4x -4x is an example of a term. x is called the variable or unknown -4 is called the coefficient Adding and Subtracting Polynomials To add and/or subtract polynomials the terms must be like terms. Like terms are terms that have the same variables with the same value of exponent for each variable. Example Underline the terms that are like terms. a) x, x, 7x, y, x b) 7xy, 5x y, y x, 5xy Copyright 004, Durham Continuing Education Page of 6

3 MPMD Principles of Mathematics Unit Lesson 6 Solution a) x, x, 7x, y, x b) 7xy, 5x y, y x, 5xy Example The order of the variables does not matter. Simplify. a) ( x ) b) ( a ) (x+ 7) + 4 (a ) c) ( 4w + 6w 8) ( 9w + w + 5) Solutions a) ( x ) (x+ 7) b) ( a ) The first set of brackets can be removed if there is no value or negative sign in front of the brackets. ( x ) (x + 7) 5x + + x + 7 5x + x x (a ) a+ 4 a+ If this is a plus sign the second set of brackets can also be removed. a a+ 4+ a + 7 Group like terms and add the like terms together. Distributive Law: a(b+c) ab +ac a(b-c)ab-ac where a, b, and c can be any real number Because this sign is negative the distributive law is used. c) ( 4w 6w 8) ( 9w w 5) w + 6w 8 9w w 5 4w 9w + 6w w 8 5 5w + 4w Copyright 004, Durham Continuing Education Page of 6

4 MPMD Principles of Mathematics Unit Lesson 6 Support Questions. State the like terms in each group. a) b) x y z x x w v,5,5,,,, 4 x, y,4 z, y, y,4w. Simplify. a) ( x+ ) + (4x+ ) b) ( 5 n) ( 6n+ ) c) d) e) f) g) h) (8a a ) ( 5a 4a 7) ( 6x 5x ) (4x 5 x) + + ( m n ) (7 6 m n ) + ( 6 x ) (7 x ) (5 6 w ) ( w ) + + (5x x) ( 4x 5 x ). Simplify. Then determine the value of the polynomial when n and when n. a) b) + (7n n 4) ( 4n n ) (n 7n ) ( n 6n ) Multiplying and Dividing Polynomials To multiply and/or divide polynomials the terms do not have to be like terms. Example Simplify. a) b) 4 (4 x )( x ) 6x y 8 4 x y c) ( w y) (5 wx) d) 00x yz 5xyz Copyright 004, Durham Continuing Education Page 4 of 6

5 MPMD Principles of Mathematics Unit Lesson 6 Solutions a) 4 (4 x )( x ) 4 x x x x x x 6 x 4 or ( 4)( )( x )( x ) x + 6 x 4 Use the multiplication of exponents rule b) 6x y 8 4 x y 6 x x x x y y y 8 x x x y - 6 x x x x y y y 8 x x x y Divide common factors x y y xy or 6 8 x y xy 4 c) ( wy ) (5 wx) 0w + yx 0w yx d) 00x yz 5xyz 4x y z 4xz Any base to the zero exponent equals Copyright 004, Durham Continuing Education Page 5 of 6

6 MPMD Principles of Mathematics Unit Lesson 6 Support Questions 4. Determine the product or quotient. a) b) c) d) e) f) g) h) i) 5 ( m )( 7 m ) ( n ) ( 6 n) 5 0 7ab 5 ba ( x )(4 xy)( x y ) 6 8a 5 7a (8 z )(7 z ) ( 4 ab) ( ab) (5 x )(4 x ) 0 ab a Simplify. Then determine the value of the polynomial when a and when b. a) b) ( ab ) ( ab )( 5 ab ) Multiplying Polynomials with a Monomial This process requires the use of the distributive law. Example Expand. a) (x 4) b) x( x 5) c) ( p )( 5 p+ p ) Copyright 004, Durham Continuing Education Page 6 of 6

7 MPMD Principles of Mathematics Unit Lesson 6 Solutions a) (x 4) 6x 8 ()(x) 6x and ()(-4) -8 b) x( x 5) 6x 0 x c) ( p )( 5 p p ) + 9p + 5p p 4 Support Questions 6. Expand. a) x(x 9) b) ( 4 n)(n ) c) b(b b+ ) d) ( x)( x ) e) f) (6 x )(7 x ) ( 4 m)( m m) Greatest Common Factor (GCF) of Polynomials Factoring means to express a polynomial as a product of factors. Expanding means to multiply factors together to form a product. Example Factor fully and check by expanding. a) x + 6 b) 4x + x 4 c) 5mn 0mn+ 0mn Copyright 004, Durham Continuing Education Page 7 of 6

8 MPMD Principles of Mathematics Unit Lesson 6 Solution Factor fully and check by expanding. a) x + 6 is largest term that divides evenly into both x and 6 x + 6 ( x + ) x x 6 Check by expanding is the Greatest Common Factor (G.C.F.) b) 4x + x ( )( x) ( )( ) x 6 x + 6 4x + x 7 x(x+ ) Check by expanding ( )( ) ( x)( ) 7x x 4x 7 x Divided both terms by the GCF which is 7x 4x + x c) 5mn 0mn+ 0mn 4 4 5mn 0mn+ 0mn 5 mnn ( m + 6 mn) Check by expanding ( )( ) 5mn n 5mn ( )( ) mn m mn ( )( ) 5mn 6mn 0mn Divided both each term by the GCF which is 5m n. 5mn 0mn+ 0mn 4 Copyright 004, Durham Continuing Education Page 8 of 6

9 MPMD Principles of Mathematics Unit Lesson 6 Support Questions 7. Factor fully. a) b) c) d) e) f) g) x 9 8n 4n b b x x + 4x 6x y x y xy x y xy 4a b + 6ab 8ab 8. Simplify each expression then factor. a) b) c) d) y y y y y y x x+ x + 6x 7 9 5t + t t + t 6w 5w + 7 w w + Key Question #6. State the like terms in each group. ( marks) a) b) 4 w,5 w,5 z, x, x, w,v 4 x, x,4 z, y, y,4w. Simplify. (8 marks) a) ( t + ) + (7t + 5) b) (6 4 r) ( 5r + ) c) d) e) f) g) h) (4n + 4n+ ) + ( 7n n 6) ( 4x + x+ ) (x + 7 x) ( + 5 m+ n ) (5 m+ 4 n ) ( + 7 x ) + (4 6 x ) ( 7 w ) ( w ) (6x 7 x) + ( x+ 9 x ) Copyright 004, Durham Continuing Education Page 9 of 6

10 MPMD Principles of Mathematics Unit Lesson 6 Key Question #6 (continued). Simplify. Then determine the value of the polynomial when n- and when n. (6 marks) a) b) (5n + n 4) + ( n + 4n ) (7n 5n ) ( n + 6n+ 8) 4. Expand. (6 marks) a) w(w + 4) b) 4 n(5n 9) c) c(7c 5c 6) d) ( h)( h+ 6) e) f) (7 t )( + t ) ( 6 x)( x + x) 5. Determine the product or quotient. (9 marks) a) b) c) d) e) f) g) h) i) 5 ( x )( 4 x ) (6 m ) ( 9 nx) 9 6 ab 5ba 5 (8 x )( xy)( xy) 7 75w 4 5w 7 ( d )(9 d ) ( ab) ( ab) (7 k )( k ) 4 ab ab Simplify. Then determine the value of the polynomial when a- and when b. (4 marks) a) b) ( ab ) ( ab)( 4 ab) Copyright 004, Durham Continuing Education Page 0 of 6

11 MPMD Principles of Mathematics Unit Lesson 6 Key Question #6 (continued) 7. Factor fully. (7 marks) a) b) c) d) e) f) g) 7y + 0x 5x 8s 7s w 6w w 8nm+ 4nm 5xy + 5x y 0xy ab 6ab + 8ab 8. Simplify each expression then factor. (6 marks) a) b) c) 7h h + h 9h + 5y + y 8 5y + 7y n n n n 9. When the terms of a polynomial in x are arranged from the highest to the lowest powers of x, the polynomial is in descending order. Simplify the following polynomial in descending order then evaluate for n ( marks) ( n n) ( n n ) ( n n ) When are the sum, difference, product and quotient of two monomials also a monomial? ( marks) Copyright 004, Durham Continuing Education Page of 6

12 Algebra Lesson 7

13 MPMD Principles of Mathematics Unit Lesson 7 Lesson Seven Concepts Introduction to algebra Solving for unknowns Checking solutions to algebraic equations Algebra Solving Equations When solving algebraic equations we must try to think of a scale always in equilibrium (balanced). It is important to keep the scale balanced at all times. What you do to one side of the equation must also be done to the other side of the equation. + + You need to get all the terms with the variable to one side and the constants (the ones with out any letters) to the other. It does not matter which side you choose for the isolating of each. Example Solve each equation algebraically. Check your solution a) 4 7k + k b) x 7 8 x c) + 4 d) 6n+ n+ Copyright 004, Durham Continuing Education Page of 6

14 MPMD Principles of Mathematics Unit Lesson 7 Solution a) -4-7k + k This side chosen for the k s -4-7k + k -4-7k - k + k - k -4-8k - k from both sides to keep scale balanced +4 from both sides to keep scale balanced k + 4 Checking the solution - 8k 6-8k k - Divide 8 from both sides to keep scale balanced Should now check the answer - 4 7k + k - 4 7(-) + (-) Substitute your answer into the original equation same If both side equal the same amount then your answer is correct. K - b) x 7 8 x x 45 x 45 x 5 Copyright 004, Durham Continuing Education Page 4 of 6

15 MPMD Principles of Mathematics Unit Lesson 7 Checking the solution x 7 8 (5) both side equal the same amount therefore the answer is correct. x c) + 4 x + 4 x + 4 x 4 + x x x + 8 x x 0 Write each term as a fraction Find the (lowest common multiple) LCM of, and 4 (the denominators) and multiply each term by the LCM. Divide the LCM by the denominator in each term then multiply this quotient by the numerator. Continue to do the same to both sides keeping the scale balanced Checking the solution Copyright 004, Durham Continuing Education Page 5 of 6

16 MPMD Principles of Mathematics Unit Lesson 7 d) 6n+ n+ 6n+ n+ 6n n + + 6n n n n n+ 6n m+ 6n+ 4 6n 6n+ 6n 6n+ 6n 6n+ 4n 4n 4 4 n 4 Find the (lowest common multiple) LCM of, and (the denominators) and multiply each term by the LCM. Divide the LCM by the denominator in each term then multiply this quotient by the numerator. Continue to do the same to both sides keeping the scale balanced Checking the solution Remember that a common denominator is needed to add fractions Copyright 004, Durham Continuing Education Page 6 of 6

17 MPMD Principles of Mathematics Unit Lesson 7 Support Questions. Solve each equation. Check your solution a) 5w 0 b) x 5 6 c) 0 t 7 n d) 8 5 e) 4w w + f) r 5 4r + 7 g) 5 6x x+ 5 h) + 7c c i) ( w + ) (5 w) j) 4( j) 7( j + 0) 4 k) h n n l) + 5 d m) 4 n) 7 ( x) bh. The formula for the area of a triangle is A. The area of a triangle is 6 cm and its base is cm. Using algebra, what is the height of the triangle?. The cost of a hall rental for a wedding is $500. Each meal at the wedding will cost an additional $ a) Write and algebraic equation for the total cost of the wedding. b) Calculate how many people attended the wedding if the wedding cost totalled $9.50. Copyright 004, Durham Continuing Education Page 7 of 6

18 MPMD Principles of Mathematics Unit Lesson 7 Key Question #7. Solve each equation. Check your solution. (4 marks) a) w 6 b) n 7 c) 5 w 45 d) 5c 7c e) 6+ 6h h 5 f).5x 4 +.x.5 g).x+.(.4 x) 4 h) x r i) 5 4 j) ( x ) k) 5( x ) 5 l) (t + 6) 0 m) ( p+ ) ( p ) n n) +. The formula for the perimeter of a rectangle with length l and width w is Pl+w. A rectangular field is 0 m long and requires 45 m of fencing to enclose it. Determine the width of the field. ( marks). Volcanoes prove that the Earth s center is hot. The formula T 0d +0 is used to estimate the temperature, T degrees Celsius, at a depth of d kilometres (km). (6 marks) a) What does each term on the right side of the equation represent? b) Estimate the depth where the temperature is 60 C. c) What is the approximate temperature at a depth of 4 km? Copyright 004, Durham Continuing Education Page 8 of 6

19 MPMD Principles of Mathematics Unit Lesson 7 Key Question #7 (continued) 4. The cost, C dollars, to produce a school yearbook is given by the equation C n, where n is the number of yearbooks printed. (8 marks) a) What does each term on the right side of the equation represent? b) Suppose there is $ 500 to spend on yearbooks. How many yearbooks can be purchased? c) How many yearbooks can be produced for $0 000? d) How much would 700 yearbooks cost? 5. Suppose you were asked to explain how to solve the equation x 5 4x+ over the phone to a friend. Explain in detail the steps that you would tell your friend to solve the equation. (4 marks) Copyright 004, Durham Continuing Education Page 9 of 6

20 Problem Solving Algebraically Lesson 8

21 MPMD Principles of Mathematics Unit Lesson 8 Lesson Eight Concepts Introduction to problem solving 5 steps suggested to problem solving Solving Problems using Algebraic Modeling Problem Solving Steps Example. Use a variable to represent the unknown quantity.. Express any other unknown quantities in terms of this variable.. Write an equation and solve. 4. Check your answer to the question. 5. State the answer to the question with a therefore statement. Members of a Girl Guide troop sold boxes of cookies to raise money for their year end camp. Brianna sold 8 more boxes than her friend Nicola. They sold a total of 46 boxes. How many boxes did each sell? Solution. Use a variable to represent the unknown quantity. Let b represent the number of boxes sold by Brianna.. Express any other unknown quantities in terms of this variable. If boxes sold by Brianna b; then boxes sold by Nicola b 8.. Write an equation and solve. b+ b b 8 46 b b 54 b 54 b 7 Copyright 004, Durham Continuing Education Page of 6

22 MPMD Principles of Mathematics Unit Lesson 8 4. Check your answer to the question. LHS RHS b+ b Example 5. State the answer to the question with a therefore statement. Brianna sold 7 boxes of cookies and Nicola sold 9. An airplane travels 7 times faster than a train. The difference in their speeds is 40 km/hr. How fast is each vehicle traveling? Solution. Use a variable to represent the unknown quantity. Let v represent the speed the train travels.. Express any other unknown quantities in terms of this variable. If the train travels at v; then the airplane travels 7v.. Write an equation and solve. 7v v 40 6v 40 6v v 70 Copyright 004, Durham Continuing Education Page of 6

23 MPMD Principles of Mathematics Unit Lesson 8 4. Check your answer to the question. LHS RHS 7v - v 40 7(70) State the answer to the question with a therefore statement. the train travels at 70 km/hr and the airplane travels at 490 km/hr. Support Questions. Find two consecutive numbers with a sum of 4.. A set of golf clubs and bag cost $5. The clubs cost $60 more than the bag. How much do the clubs cost?. Brianna and Noah ran as far as they could in 60 min. Brianna ran.5 km less than Noah. Together they ran 7.5 km in total. How far did each run. 4. A bottle and a cork cost $.0. The bottle cost $0.90 more than the cork. How much does the cork cost? 5. Don and his brother Dan went fishing. Don caught 6 times more mass of fish than his brother. Together they caught 4. kg of fish. What mass of fish did each brother catch? Copyright 004, Durham Continuing Education Page of 6

24 MPMD Principles of Mathematics Unit Lesson 8 Key Question #8. The combined mass of two children is 75 lbs. The first child is four times the mass of the second child. What are the masses of the two children? ( marks). pieces of wood have a total length of m. The second piece is twice the size of the first and the third piece is more metres than triple the first. What are the lengths of the pieces of wood? (4 marks). Abdul drove four trips that totalled 46 km. The second trip was 4 less than double the first. The third trip was double the second and the fourth trip was 8 more than triple the first. How far was each trip? ( marks) 4. A rectangle has a width that is cm less than its length. The perimeter of the rectangle is cm. What is the length and width of the rectangle? ( marks) 5. The same numbers of each coin have a total of $6.00. There are nickels, dimes and quarters. How many of each coin are there? (4 marks) Copyright 004, Durham Continuing Education Page 4 of 6

25 Slope Lesson 9

26 MPMD Principles of Mathematics Unit Lesson 9 Lesson Nine Concepts Introduction to slope Cartesian plane x and y coordinates on the Cartesian plane Plotting order pairs Quadrants of the Cartesian plane Recognizing positive, negative, zero and undefined slopes Using the rise and the run of a given line to find its slope Using a pair of coordinates of a line to calculate slope Cartesian Plane Cartesian Plane uses the x and y axes to plot a point identified by a pair of numbers. If it is known that a point has an x-coordinate of, then this point could be located anywhere along the vertical line passing through on the x-axis. If it is known that a point has a y-coordinate of 5, then this point could be located anywhere along the horizontal line passing through 5 on the y-axis. The point of intersection of the lines formed by x - and y 5 is the location of a point given by the ordered pair (-, 5). x-coordinate y-coordinate This grid system is the Cartesian plane Vertical axis is the y-axes The Cartesian plane is divided into four quadrants A Horizontal axis is the x-axes Copyright 004, Durham Continuing Education Page 6 of 6

27 MPMD Principles of Mathematics Unit Lesson 9 Example Plot and label each point: A(,5), B(0,-6), C(-7,-4) Solution Support Questions. Properly draw and label a Cartesian plane then plot and label the following ordered pairs. A(, ), B(, 5), C(, -4), D(7, 0), E(0, 8), F(-4, -8), G(-5, 9), H(-, 0). State the quadrant for each of the ordered pairs plotted in question one.. The points (-, 5) and (, 5) are two vertices of a square. State all other order pairs that could be the other two vertices of the square. Copyright 004, Durham Continuing Education Page 7 of 6

28 MPMD Principles of Mathematics Unit Lesson 9 Slope Slope describes the steepness of a line or line segment; the ratio of the rise of a line or line segment to its run. Slope is either one of the following four types: a) positive b) negative c) no slope d) undefined The formula for determining slope is: Example rise slope run. a) Draw a board that would lie on the staircase b) State the slope of the staircase Solution + + rise slope run + slope + Copyright 004, Durham Continuing Education Page 8 of 6

29 MPMD Principles of Mathematics Unit Lesson 9 Example c) State the slope of the hypotenuse of the triangle given below. 4 Solution 4 rise slope run + 4 slope slope Reduced to simplest form. Support Questions 4. Find the slope of each staircase. a) b) Copyright 004, Durham Continuing Education Page 9 of 6

30 MPMD Principles of Mathematics Unit Lesson 9 Support Questions 5. This drawing represents the side view of road passing through a mountain range. The road moves from left to right. Calculate the slope of each line segment. 6. The slope of a line segment is 6. What is a possible rise and run? 7. The slope of a line segment is 4. What is the rise if the run is? 8. State the slope of each line segment. Copyright 004, Durham Continuing Education Page 0 of 6

31 MPMD Principles of Mathematics Unit Lesson 9 Slope between two points on a Cartesian Plane The slope of line segment can be determined if two ordered pairs of the line segment are known. y y The following formula is used: slope x xz Example a) Find the slope of the line containing the coordinate pairs A(7,) and B(-9,7). b) Find the slope of the line containing the coordinate pairs A(-,) and B(4,6). Solutions ( x, y ) ( x, y ) The s in the first ordered pair denotes the x and y values of that term and the s in the second ordered pair denotes the x and y values of that term. a) (7, ) and (-9, 7) slope y x y (7) () x ( 9) (7) Substitute the values into the equation for slope. 4 6 Fraction needs to be simplified. 4 Therefore the slope is b) (-,) and (4,6) slope y y x x. 4 (6) () (4) ( ) Therefore the slope is 5 7. Copyright 004, Durham Continuing Education Page of 6

32 MPMD Principles of Mathematics Unit Lesson 9 Support Questions 9. Find the slope of the line containing the ordered pairs. a) A(4,-5), B(-4,6) b) C(,8), D(7,) c) E(0,), F(,) d) G(5,4), H(,-) e) I(,), J(-6,-) f) L(,7), M(,-) g) N(4,-6),P(,-6) 0. Find the missing value in the ordered pair. a) (,-), (x,6); slope -4 b) (,), (7,y); slope c) (x,4), (,-); slope -5 d) (8,y), (-,-) 5 Key Question #9. Properly draw and label a Cartesian plane then plot and label the following ordered pairs. (6 marks) A(, ), B(, -5), C(, 4), D(-7, 0), E(0, -8), F(, -8), G(-4, -9), H(, 0). State the quadrant for each of the ordered pairs plotted in question one. ( marks). The points (-4, ) and (4, ) are two vertices of a square. State all other order pairs that could be the other two vertices of the square. (4 marks) Copyright 004, Durham Continuing Education Page of 6

33 MPMD Principles of Mathematics Unit Lesson 9 4. This drawing represents the side view of road passing through a mountain range. The road moves from left to right. Calculate the slope of each line segment. (6 marks) 5. The slope of a line segment is. What is a possible rise and run? ( marks) 6. The slope of a line segment is -5. What is the rise if the run is 4? ( marks) 7. State the slope of each line segment. (6 marks) 8. Find the slope of the in line containing the ordered pairs. (7 marks) a. A(,-5), B(-,6) b. C(,6), D(5,) c. E(,), F(,) d. G(-,4), H(,-) e. I(,), J(-6,) f. L(4,5), M(4,-7) g. N(,-5),P(,-8) Copyright 004, Durham Continuing Education Page of 6

34 MPMD Principles of Mathematics Unit Lesson 9 Key Question #9 (continued) 9. Find the missing value in the ordered pair. (4 marks) a) (-,-4), (0,y); slope b) (-,), (x,5); slope 7 c) (x,5), (,-5); slope 5 e) (-6,y), (8,) 7 0. Decide if the following statements are true, sometimes true or not true. Explain. (8 marks) a) An ordered pair with positive co-ordinate and negative coordinate lies in the rd quadrant. b) An ordered pair with both co-ordinates positive coordinates lies in the st quadrant. c) An ordered pair where the x and y coordinate are the same lies in the st or rd quadrant. d) The order pairs on a vertical line have the same y coordinate. Copyright 004, Durham Continuing Education Page 4 of 6

35 Relationships In Data Lesson 0

36 MPMD Principles of Mathematics Unit Lesson 0 Lesson Ten Concepts Introduction to relationships in data Graphing relationship in data Working with table of values Creating graphs for tables of values Using graphs to solve related questions Relationships in Data Tables and graphs of data help to show the relationships between quantities. In mathematics the relationship between a pair of quantities is called a relation. Example Use the graph below to answer each question. a) State the percentage of Canadians who enjoy professional wrestling for each of the following years. Show the popularity in a table b) When did the popularity of professional wrestling reach 6% of all Canadians? Copyright 004, Durham Continuing Education Page 6 of 6

37 MPMD Principles of Mathematics Unit Lesson 0 Solution a) State the percentage of Canadians who enjoy professional wrestling for each of the following years? Show the popularity in a table Year Popularity (%) b) When did the popularity of professional wrestling reach 6% of Canadians? Approximately 984 Example Which graph below best represents each situation? a) the height of a person over time b) the height of roller coaster over time c) the amount of hours of sunlight over a year d) the number of D.V.D. players sold compared to selling price i) ii) iii) iv) Copyright 004, Durham Continuing Education Page 7 of 6

38 MPMD Principles of Mathematics Unit Lesson 0 Solution Which graph below best represents each situation? a) the height of a person over time b) the height of roller coaster over time c) the amount of hours of sunlight over a year d) the number of D.V.D. players sold compared to selling price i) ii) iii) iv) a d c b Support Questions. Use the graph below to answer each question. a) What was Canada s population in the following years? b) When did Canada s population reach 5 million? Canadians Population Since Population (millions) Year Copyright 004, Durham Continuing Education Page 8 of 6

39 MPMD Principles of Mathematics Unit Lesson 0 Support Questions. Brianna walks to her grandparents. This graph shows her distance from home during one of her walks. Describe her walk.. Noah is riding his bike from his grandparents to his home. Describe Noah s possible ride home 4. Refer to the graph given below: a) How many textbooks make the following heights? 45 cm, 60cm, 75cm b) Approximately how high are the following stacks of textbooks? 7 books, 9 books, 4 books, books Copyright 004, Durham Continuing Education Page 9 of 6

40 MPMD Principles of Mathematics Unit Lesson 0 Support Questions 5. Construct a graph of these data. Number of stairs climbed Height (cm) a) Did you join the points? Explain. b) What is the height of 5 stairs? c) How many stairs will reach a height of 444 cm? d) If the number of stairs is doubled will the height double? Explain. 6. Construct a graph of these data. Radius of a circle, (cm) Area of a circle. ( cm ) a) Did you join the points? Explain. b) What is the approximate radius of circle with an area of 00cm? c) What is the approximate area of circle with a radius of 4.5 cm? d) If the radius is doubled will the area double? Explain. Copyright 004, Durham Continuing Education Page 40 of 6

41 MPMD Principles of Mathematics Unit Lesson 0 Graphing Relations Relations can be either linear or non-linear. Linear means the relation forms a single straight line and non-linear produces anything that is not a single straight line. y y The following formula is used: slope x xz Example Draw a graph of the relation described by the equation. a) y x+ b) y n Solutions a) y x+ First step is to make a table of values and choose values to place in x column of table x y Second step is to substitute each value into the equation to determine the y value. For x - y x+ (-) x y Copyright 004, Durham Continuing Education Page 4 of 6

42 MPMD Principles of Mathematics Unit Lesson 0 For x - y x+ (-)+ + 4 x For x 0 y y x+ (0)+ 0 + x y For x y x+ () x y Copyright 004, Durham Continuing Education Page 4 of 6

43 MPMD Principles of Mathematics Unit Lesson 0 For x y x+ () x y Next, plot the coordinates (x, y) on a grid and join the points with a straight edge and label the equation. Copyright 004, Durham Continuing Education Page 4 of 6

44 MPMD Principles of Mathematics Unit Lesson 0 b) y n x y y n Support Questions 7. Complete the table of values. a) y x 5 b) y x+ 6 c) y w Copyright 004, Durham Continuing Education Page 44 of 6

45 MPMD Principles of Mathematics Unit Lesson 0 Support Questions 8. The cost of D dollars, to print and bind y copies of a yearbook is given by the equation D 60 +0n. a) Make a table of values to show the costs for up to 400 Yearbooks. b) Use the table to draw a graph. c) Use the graph to estimate the cost of 5 copies. d) Use the graph to estimate how many copies can be made for $ Complete a table of values and graph each relation. a) t s + 5 b) y n 5 c) p 5.5 w.5 0. Which of the following ordered pairs satisfy the relation modeled by Show by substituting the values into the relation. a) (-, 6) b) (4,5) c) (7,.5) d) (-6, 0) e) (-4, 6.5) y x+ 7. Key Question #0. Which graph below best represents each situation? ( marks) a) the height of a person over time b) the height of roller coaster over time c) the amount of hours of sunlight over a year d) the number of D.V.D. players sold compared to selling price i) ii) iii) iv). Complete a table of values and graph each relation. ( marks) a) C n+ 0 b) y x 4 5 c) W 4.75n 0.50 d) P l + e) m n + f) b a + 4 Copyright 004, Durham Continuing Education Page 45 of 6

46 MPMD Principles of Mathematics Unit Lesson 0 Key Question #0 (continued). The time that passes between the time you see lightning and you hear the thunder depends on your distance from the lightning. With each km from the lightning seconds pass. (5 marks) a) Make a table of values for distances from 0 to 5 km. b) Graph this relation. State whether it is a linear or non-linear relation. c) Using your graphed relation, how much time passes before you hear lightning that occurs 4.5 km away? d) Using your graphed relation, how far away are you if you hear the thunder in.5 seconds? 4. The amount a taxi driver charges a costumer is given by the equation A.5k + 5.5, where A is the total amount charged and k is the kilometres driven. (6 marks) a) What do the numbers in the equation represent? b) Make a table of values for distances from 0 to 0 km. c) Graph this relation. State whether it is a linear or non-linear relation. d) Using your graphed relation, how much is charged if a person goes 7.5 km? e) Using your graphed relation, how far can a person go in a taxi for $5? 5. A sales person earns a monthly salary of $500 and 8% commission on all sales for that month. (6 marks) a) Write an equation to model this relation. b) Make a table of values. c) Graph this relation. State whether it is a linear or non-linear relation. d) Using your graphed relation, what were the sales if the sales persons total earnings for the month were $800? e) Using your graphed relation, how much did the sales person earn if there was $50 in sales? 6. Ashlee repairs DVD players. She charges $5 to inspect the problem and $0/h to repair the device. (5 marks) a) Write an equation to model this relation. b) Make a table of values. c) Graph this relation. d) Using your graphed relation, how long did it take Ashlee to repair the DVD player if she charged $55? e) Using your graphed relation, how much should Ashlee charge if it takes her 4.5 hours to repair the DVD player? Copyright 004, Durham Continuing Education Page 46 of 6

47 MPMD Principles of Mathematics Unit - Support Question Answers Support Question Answers Lesson 6. a) x, x b) y, y. a) b) ( x+ ) + (4x+ ) x+ + 4x+ x+ 4x+ + x + 5 ( 5 n) ( 6n+ ) 5n+ 6n 5n+ 6n+ n + c) d) e) f) g) a + a + a + a+ (8 ) ( 5 4 7) a + a a + a a a + a+ a a + a ( 6x 5x ) (4x 5 x) 6x + 5x+ 4x 5+ x x x + x+ x x + x 0 7 ( ) (7 6 ) m n + m+ n m n + 7 6m+ n 0 8m ( 6 x ) (7 x ) + + 6x 7+ x 7+ 6x + x 5+ 9x (5 6 ) ( ) w w 5 6w + w 5 6w + w 5w Copyright 004, Durham Continuing Education Page 47 of 6

48 MPMD Principles of Mathematics Unit - Support Question Answers h) (5x x) ( 4x 5 x ) + + 5x x 4x+ 5x 5x + 5x x 4x 0x 7 x. a) b) (7n n+ 4) ( 4n n ) for n n n+ + n + n n + n n+ n+ + 7 n n+ 7 () () 7 n n for n 7 ( ) ( ) 7 n n n n for n (n 7n+ ) + ( n + 6n+ ) n 7n n 6n n n 7n 6n n n n+ 5 () () for n n+ 5 ( ) ( ) a) b) c) 5 ( m )( 7 m ) ( n ) ( 6 n) m 7 n d) e) f) ( x) (4 xy)( x y ) 6 8a 6 5 4xy 7a 4a g) h) i) ( 4 ab) ( ab) (5 x )(4 x ) a 0x 5 7ab ba ab 5 (8 z )(7 z ) 56z 4 0 ab 5 a 9 0 ab 45 ab Copyright 004, Durham Continuing Education Page 48 of 6

49 MPMD Principles of Mathematics Unit - Support Question Answers 5. a) b) ( ab) ab ( )( ) 4ab 4 6 for a, b 4ab 4() ( ) ab 4(6)() a) b) x(x 9) x 9x c) d) b(b b+ ) b b + b e) f) 4 (6 x )(7 x ) 4x 8x ( )( 5 ) ab 5ab 4 for a, b 5ab 5() ( ) 4 4 (60)() 60 ab ( 4 )( ) 8 + n n n n ( )( ) + x x x x ( 4 m)( m m) 4m + 4m 7. a) ( x ) b) 4 n(n ) c) b(b ) d) xx ( x+ 4) e) x y( xy) f) xy( y + x ) g) 4 ab( a 4b + ) 8. a) b) c) 5 y y + y y + y + y y + y + y ( 4 7) y y + y x x x x 4x 4 4( x ) t + t + + t + t t + t + 7( ) t + t + Copyright 004, Durham Continuing Education Page 49 of 6

50 MPMD Principles of Mathematics Unit - Support Question Answers d) w w + w w w w w + ( 5) w w w + Lesson 7. a) b) c) x 5 6 5w 0 x w 0 x 5 5 x w 6 x 7 0 t t 7 0 t 7 t 7 t 7 d) e) n 8 5 n 5 5(8) 5 n 90 f) g) r 5 4r + 7 r r r 4r + r 4r 4r 4r + 8r 8r 8 8 r 4 4w w + 4w + w + + 4w w + 5 4w w w w + 5 w 5 w 5 5 w 5 6x x x 6x x 6x x x x 0 x Copyright 004, Durham Continuing Education Page 50 of 6

51 MPMD Principles of Mathematics Unit - Support Question Answers h) i) + 7c c + 7c c 7c c 6 7c c c c 6 5c 6 5c c 5 j) k) 4( j) 7( j + 0) 4 8j 4j j + 8j 4j + 8j j j j 66 j j l) m) n n + 5 n n 5 5() n 0+ 5n n 5n 0+ 5n 5n n 0 n 0 n 5 ( w + ) (5 w) w w w w w w w + w w + w 5w 5w 5 5 w 5 4 h 4h 4h () 4h 6 4h h 9 d 4 d () 4 d 8 4 d d d d Copyright 004, Durham Continuing Education Page 5 of 6

52 MPMD Principles of Mathematics Unit - Support Question Answers n) 7 ( x) 7 6+ x x x x x. bh A ()( h) 6 h (6) 5 h 5 h h Therefore the height of the triangle is cm.. a) C 45.50m+500 b) m m m m m Therefore 75 people would attend. Copyright 004, Durham Continuing Education Page 5 of 6

53 MPMD Principles of Mathematics Unit - Support Question Answers Lesson 8. Let n be the first consecutive number. If the first consecutive number is n then the next number is n+ n+ n+ 4 n + 4 n + 4 n 4 n 4 n 7 the consecutive numbers are 7 and 7. Let b be the cost of the bag If b is the cost of the bag then the clubs are b+60 b+ b b b b 65 b 65 b 8.5 the bag cost $8.50 and the golf clubs cost $4.50. Copyright 004, Durham Continuing Education Page 5 of 6

54 MPMD Principles of Mathematics Unit - Support Question Answers. Let b be the distance ran by Brianna If b is the distance ran by Brianna then Noah ran b+.5 b+ b b b b 5 b 5 b 7.5 Brianna ran 7.5 km and Noah ran 0 km. 4. Let c be the cost of the cork If c is the cost of the bottle then costs c c + c c c c.0 c.0 c.0 the cork cost $ Let d be the mass of the fish Dan caught. If d is the mass of the fish Dan caught then Don s fish s mass is 6d. 6d + d 4. 7d 4. 7d d.6 Don caught.6 kg of fish and Dan caught.6 kg of fish. Copyright 004, Durham Continuing Education Page 54 of 6

55 MPMD Principles of Mathematics Unit - Support Question Answers Lesson 9.. a) I b) I c) IV d) I and IV e) I and II f) III g) II h) II and III. 4. a) rise + run + b) rise - run + rise m run rise m run + Copyright 004, Durham Continuing Education Page 55 of 6

56 MPMD Principles of Mathematics Unit - Support Question Answers 5. AB slope rise m run BC slope rise m run CD slope rise 0 m 0 run 4 DE slope rise m run + EF slope rise 0 m 0 run FG slope rise m run GH slope rise 0 m 0 run HI slope rise m run 6. Answers may vary. 8 6 so the a possible rise is 8 and a possible run is x 4 x (4) x Therefore the rise is +. Copyright 004, Durham Continuing Education Page 56 of 6

57 MPMD Principles of Mathematics Unit - Support Question Answers a) rise slope m run b) rise 0 slope m 0 run c) rise slope m run 8 d) rise slope m run + e) rise slope m undefined run 0 f) rise slope m run a) b) c) y y (6) ( 5) slope m x x ( 4) (4) 8 8 y y () (8) 6 slope m x x (7) () 4 y y () () slope m x x () (0) d) slope m y y ( ) (4) 7 7 x x () (5) 4 4 e) ) y y ( ) () 5 5 slope m x x ( 6) () 8 8 y y ( ) (7) 9 x x () () 0 f slope m undefined y y ( 6) ( 6) 0 g slope m x x () (4) ) 0 Copyright 004, Durham Continuing Education Page 57 of 6

58 MPMD Principles of Mathematics Unit - Support Question Answers 0. ) y y (6) ( ) 8 4 ( ) () a slope m x x x x (8)() ( 4)( x ) 8 4x + 8 4x + 4 4x 4 4x 4 4 x y y ( y) () y b) slope m x x (7) () 6 ) (6)() ()( y ) y + y + y y y ( ) (4) 5 5 () ( ) c slope m x x x x ( 5)() ( 5)( x) x x 0 5x 0 5x 5 5 x y y ( ) ( y) y d) slope m x x ( ) (8) 0 5 ( 0)() (5)( y) 0 5 5y y 5 5y 5 5y y Copyright 004, Durham Continuing Education Page 58 of 6

59 MPMD Principles of Mathematics Unit - Support Question Answers Lesson 0. a) 6 million; 9 million; million; 6 million; 9 million b) 98 Canadians Population Since Population (millions) Year. Walked at a steady rate, then took a brief rest then walked slightly slower but steadily then reached her grandparents and finally walked quickly back to home.. Noah was riding his bike up a hill then once he reached the top rode his bike quickly down the hill. 4. a) 6,8 and 0 textbooks b) approx 48, 68, 0 and 5 cm. 5. Walk Those Stairs Height (cm) Number of Stairs Copyright 004, Durham Continuing Education Page 59 of 6

60 MPMD Principles of Mathematics Unit - Support Question Answers a) Yes, points are joined to see the possible values between points. b) 90 cm c) stairs d) yes because this is a linear relationship. 6. Circles 00 Area of Circle Radius of Circle (cm) a) Yes, points are joined to see the possible values between points. b) 5.4 cm c) 6cm d) No, this is not a linear relationship. 7. a) y x 5 b) y x + 6 c) y w X x -5 Y -4 (-4) (-) -5-7 () -5-5 (5) -5 5 X -x + 6 Y - -(-) (-) (0) () X w^ - Y - (-)^ (0)^ - - ()^ - 0 ()^ - Copyright 004, Durham Continuing Education Page 60 of 6

61 MPMD Principles of Mathematics Unit - Support Question Answers 8. X Y b) 9. a) c) $00 d) 00 yearbooks Copyright 004, Durham Continuing Education Page 6 of 6

62 MPMD Principles of Mathematics Unit - Support Question Answers b) c) 0. c) (7,.5) b) (4, 5) y x+ 7 5 ( 4) (7) e) (-6, 0) 0 ( 6) Copyright 004, Durham Continuing Education Page 6 of 6