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MAP4C Foundations for College Mathematics, Grade 12, College Preparation Lesson 6 Exponents MAP4C – Foundations for College Mathematics Lesson 6 Lesson Six Concepts ¾ Evaluate simple numerical expressions involving rational exponents, without using technology ¾ Evaluate numerical expressions involving negative exponents, with and without using scientific calculators ¾ Simplify algebraic expressions involving integral exponents, using the laws of exponents Law #1: Multiplying Powers When multiplying variables with the same base, we add the exponents together. Example 1: Simplify the following: a. 53 x 57 b. (-3)2 x (-3) c. (x-3)(x11) b. (-3)2 + 1 = (-3)3 c. x(-3) + 11 = x8 Solution a. 53 + 7 = 510 Law #2: Dividing Powers When dividing variables with the same base, we subtract the exponents. Example 2: Simplify the following: a. 12 5 12 3 b. Solution a. 125 - 3 = 122 y 15 y 14 c. ( −6) 9 ÷ ( −6) −4 b. y15 – 14 = y1 or y c. (-6)9 – (-4) = (-6)13 Law #3: Power to a Power When we have a power to a power, we multiply the exponents together. Example 3 Simplify the following: a. (64)7 b. [(-2)5]4 Copyright © 2007, Durham Continuing Education c. (s2)2 Page 2 of 45 MAP4C – Foundations for College Mathematics Lesson 6 Solution a. 64 x 7 = 628 b. (-2)5 x 4 = (-2)20 c. s2 x 2 = s4 Zero Exponents Anything with an exponent of 0 equals 1. ***The Exception is: if the base is 0, then the number cannot be solved Example 4 Simplify: a. 20 b. (-4)0 ⎛3⎞ c. ⎜ ⎟ ⎝5⎠ b. 1 c. 1 0 d. 00 Solution a. 1 d. cannot be solved Negative Exponents Anything with a negative exponent gets flipped and the negative exponent becomes positive (this is called taking the negative reciprocal). Simplify will always mean to put the power into a single positive exponent. Example 5 Simplify: -1 1 21 a. 2 ⎛3⎞ d. ⎜ ⎟ ⎝4⎠ −2 b. 4 1 c. −6 3 ⎛ 1⎞ b. ⎜ 3 ⎟ ⎝4 ⎠ ⎛4⎞ d. ⎜ ⎟ ⎝3⎠ 2 c. 36 -3 Solution a. Example 6 Simplify: a. n 4 × n −5 n6 b. (2a2)4 x 3(a4)2 Copyright © 2007, Durham Continuing Education c. (5x 3 )( y 4 − 4x 2 y 6 (2xy)2 ) Page 3 of 45 MAP4C – Foundations for College Mathematics Lesson 6 Solution a. n4 + (-5) – 6 = n-7 1 n7 = b. (24a2 x 4) x 3(a4 x 2) c. 5( −4)x 3 + 2 y 4 + 6 22 x1x2 y1x2 − 20x 5 y 10 4x 2 y 2 = (16a8)(3a8) = = 16(3)a8 + 8 = −5x 5 −2 y10 −2 = 48a16 = −5x 3 y 8 Rational Exponents Often values can have exponents that are fractions. Below are examples explaining how this works. Example 7 Simplify: 8 The numerator in the fraction acts the same as any other power that is not in fraction form. 2 3 so the above is saying 82 = 64 The denominator in the fraction acts the root. 3 64 = 4 therefore 2 3 8 =4 Example 8 Simplify a. 9 1 2 b. 4 3 2 c. 9 Copyright © 2007, Durham Continuing Education − 1 2 d. 8 − 2 3 Page 4 of 45 MAP4C – Foundations for College Mathematics Lesson 6 Solution 1 2 a. 9 = 2 91 = 2 9 = ±3 3 2 b. 4 = 2 4 3 = 2 64 = ±8 c. 9 d. 8 − − 1 2 1 = 2 3 = 1 2 9 1 8 2 3 = = 1 2 9 1 1 3 8 2 =± 1 3 = 1 3 64 =± 1 4 Support Questions 1. 2. Evaluate: a. (-11)0 b. 52 c. 81-1 d. 154320 e. 6-3 f. 2-3 x (16-7)0 g. 7-2 h. (-5)2 b. (-21)-2 x (-21)-3 c. (m-3)6 Simplify: −6 ⎛2⎞ ⎛ 2⎞ a. ⎜ ⎟ × ⎜ ⎟ ⎝5⎠ ⎝5⎠ 2 2 d. (3ab c) 3. ⎛ a −2 e. ⎜⎜ −2 ⎝a ⎞ ⎟⎟ ⎠ −2 (7m n ) (− 3m n ) (3mn )(− m n ) 2 f. −5 2 4 −2 8 −4 Evaluate. a. 25 3 2 − e. 81 4. −3 b. 36 3 4 f. 16 − 3 2 c. 16 5 4 3 4 d. 8 1 g. 5 2 + 25 2 − 1 4 3 4 h. 2 5 × 2 5 Simplify 1 1 a. (15625b 6 ) 6 b. (−8b 6 c 3 ) 3 Copyright © 2007, Durham Continuing Education Page 5 of 45 MAP4C – Foundations for College Mathematics Lesson 6 Key Question #6 1. 2. Evaluate: a. (5)3 b. 70 c. 4-2 d. 10 0000 e. 5-4 f. 3-2 x (8-5)0 g. 8-1 h. (-2)5 b. 5-4 x 5-3 c. (nw2)-4 Simplify: ⎛2⎞ a. ⎜ ⎟ ⎝3⎠ −3 ⎛2⎞ ÷⎜ ⎟ ⎝3⎠ ⎛ w1 e. ⎜⎜ −3 ⎝w 3 d. (3abc) 3. 1 1 a. 49 2 b. 64 3 3 2 e. 8 −4 4 x 3 y −2 f. 2x −1y 4 c. 16 − 1 4 1 ⎛ 1 ⎞ f. ⎜⎜16 4 + 9 2 ⎟⎟ ⎝ ⎠ 5 3 3 Simplify 2 a. (64n 9 ) 3 5. ⎞ ⎟⎟ ⎠ Evaluate. d. 36 4. −5 1 b. (5b 2 c 4 ) 3 1 3 The side length, s, of a cube is related to its volume, V, by the formula s = V . A cube box when filled with materials has a volume of 729 cm 2 . What is the side length of the cube box used? Copyright © 2007, Durham Continuing Education Page 6 of 45 MAP4C Foundations for College Mathematics, Grade 12, College Preparation Lesson 7 Linear, Quadratic and Exponential Functions MAP4C – Foundations for College Mathematics Lesson 7 Lesson Seven Concepts ¾ ¾ ¾ ¾ ¾ Comparing linear, quadratic and exponential functions Understanding and graphing the y-intercept form of the line Understanding the meaning of “a” in the equation y = a( x − h)2 + k Understanding the meaning of “h” and “k” in the equation y = a( x − h)2 + k Finding the axis of symmetry given the an equation in vertex form y = a( x − h)2 + k ¾ Plotting and graphing quadratic equations ¾ Substitution into quadratic equations ¾ Introduction to graphing exponential functions Linear Function The equation of a line has 2 basic forms. One being the y-intercept form and the other is called standard form. y-intercept form takes the form y = mx + b y-intercept form It is called y-intercept form because we can use the y-intercept in the equation to help us understand and graph the equation. y = mx + b “b’s” value is the spot on the y axis where the line intercepts the y axis. “m” value is the slope of the line. Example 1 State the slope and y-intercept for the following equations. a. y = 3 x + 6 1 b. y = − x − 1 2 Copyright © 2007, Durham Continuing Education Page 8 of 45 MAP4C – Foundations for College Mathematics Lesson 7 Solution a. y = 3 x + 6 slope = m = rise 3 = =3 run 1 1 b. y = − x − 1 2 rise − 1 slope = m = = run 2 y-intercept = b = +6 y-intercept = b = -1 Example 2 Graph the following equations using the slope and y-intercept. a. y = −2x + 3 Solution a. y = −2x + 3 m = -2 = b. y = 2 x −1 3 rise − 2 = and b = +3 run 1 First plot the yintercept. Starting at the y-intercept go down 2 and right 1 then down 2 again and right one again and so on…Then draw a line through the points plotted to create your line. y = −2x + 3 Copyright © 2007, Durham Continuing Education Page 9 of 45 MAP4C – Foundations for College Mathematics b. y = Lesson 7 2 x −1 3 y= 2 x −1 3 Example 3 Find the point of intersection of these two equations. (Solve the system of equations graphically) y = −2x + 1 y = x+7 Solution The point of intersection is (-2, 5) which solves the system of equations. Copyright © 2007, Durham Continuing Education Page 10 of 45 MAP4C – Foundations for College Mathematics Lesson 7 (-2, 5) is the only two ordered pairs that will satisfy both equations. y = −2x + 1 5 = −2( −2) + 1 5=5 y = x+7 5 = −2 + 7 5=5 Support Questions 1. State the slope and y-intercept of the equations below, then properly graph and label the equations using the slope and y-intercept. a. y = −4 x + 3 2. b. y = x − 4 2 c. y = − x + 7 3 d. y = 3 x −1 5 Solve for the system of equations by graphing. (Find the point of intersection) a. y = −3 x + 4 y = −2x + 5 2 x+6 b. 3 y =x+4 y= c. y = x −1 y = 4x + 2 Quadratic Functions Opening of a Parabola y = a( x − h) + k 2 When the equation of a parabola has its “a” value as positive then the parabola opens up. Copyright © 2007, Durham Continuing Education Page 11 of 45 MAP4C – Foundations for College Mathematics Lesson 7 When the equation of a parabola has its “a” value as negative then the parabola opens down. Example 1 Using the value of “a” in the equation given, state whether its parabola would open up or down. y = −2( x − 3)2 + 5 Example 2 Using the value of “a” in the equation given, state whether its parabola would open up or down. y = ( x − 3) 2 + 5 Copyright © 2007, Durham Continuing Education Page 12 of 45 MAP4C – Foundations for College Mathematics Lesson 7 Support Questions 3. For each of the following equations state whether the parabola would open up or down. a. y = 3( x − 1) 2 + 2 d. y = − x 2 b. y = −( x − 3) 2 e. y = −1( x − 2) 2 + 1 c. y = x 2 + 5 f. y = 3( x − 4) 2 − 2 Vertex of a Parabola y = a( x − h) + k 2 When an equation of a parabola is in vertex form y = a( x − h)2 + k the values of h and k become the coordinates for the vertex of the equation. Example 1 State the coordinates of the vertex of the parabola given by the following equation: y = −2( x − 3)2 + 5 Example 2 State the coordinates of the vertex of the parabola given by the following equation: y = ( x + 2) 2 − 4 Copyright © 2007, Durham Continuing Education Page 13 of 45 MAP4C – Foundations for College Mathematics Lesson 7 Support Questions 4. For each of the following equations state the coordinates of the vertex for its parabola. a. y = 3( x − 1) 2 + 2 d. y = − x 2 b. y = −( x − 3) 2 e. y = −1( x − 2) 2 + 1 c. y = x 2 + 5 f. y = 3( x − 4) 2 − 2 Equation of the Axis of Symmetry for a Parabola As stated in the previous lesson the equation of the axis of symmetry is the equation for the vertical line that divides the parabola exactly in half vertically (up and down). The equation is always the “x” value in the vertex coordinate or represented by the “h” in the vertex form of the equation y = a( x − h)2 + k . Example 1 What is the equation of the axis of symmetry for the following equation? y = −2( x − 3) 2 + 5 Copyright © 2007, Durham Continuing Education Page 14 of 45 MAP4C – Foundations for College Mathematics Lesson 7 Example 2 What is the equation of the axis of symmetry for the following equation? y = −x 2 Support Questions 5. For each of the following equations state the equation of the axis of symmetry for its parabola. a. y = 3( x − 1) 2 + 2 b. y = −( x − 3) 2 c. y = − x 2 + 5 Graphing of a Parabola y = a( x − h) + k using a table of values. 2 The key to completing a table of values and plotting its coordinates is doing the order of operations correctly. BEDMAS (Brackets, Exponents, Division, Multiplication, Addition, Subtraction) Do the brackets first Then exponents Then any multiplication and/or division Then any addition and/or subtraction Copyright © 2007, Durham Continuing Education Page 15 of 45 MAP4C – Foundations for College Mathematics Lesson 7 Example 1 Complete the table of values below and then graph. y = ( x − 1) 2 + 2 Solution Copyright © 2007, Durham Continuing Education Page 16 of 45 MAP4C – Foundations for College Mathematics Lesson 7 Support Questions 6. For each of the following equations complete a table of values for x values from – 2 to 2 and graph. a. y = 3( x − 1) 2 + 2 b. y = −( x − 3) 2 c. y = − x 2 + 5 Exponential Functions Exponential functions were used in the finance section of this course. An exponential function results in either a rapid increase or decrease of the initial value. Example 1 Graph the exponential function y = 2 x for integer values of x from 0 to 5. Copyright © 2007, Durham Continuing Education Page 17 of 45 MAP4C – Foundations for College Mathematics Lesson 7 Solution x y 0 20 = 1 1 21 = 2 2 22 = 4 3 23 = 8 4 2 4 = 16 5 2 5 = 32 Support Questions 7. Which of the following represent exponential functions? Explain. a. y = 3 x c. 4 x 2 − 3 x + 3 b. y = 3 x d. y = 1 x+4 2 e. y = 4(5) n 8. Graph the exponential function y = 2(3) x where 0 ≤ x ≤ 5 . Key Question #7 1. State the slope and y-intercept of the equations below, then properly graph and label the equations using the slope and y-intercept. a. y = 3 x − 2 2. b. y = − x + 4 1 c. y = − x + 1 2 d. y = 2 x −3 3 Solve for the system of equations by graphing. (Find the point of intersection) a. y =x+4 y = −x + 5 1 x −1 b. 2 y =x+4 y= Copyright © 2007, Durham Continuing Education c. y = x +1 y = 2x − 5 Page 18 of 45 MAP4C – Foundations for College Mathematics Lesson 7 Key Question #7 (continued) 3. For each of the following equations state whether the parabola would open up or down. a. y = −2( x + 1) 2 + 3 4. b. y = ( x − 1) 2 + 2 c. y = − x 2 + 3 For each of the following equations state the equation of the axis of symmetry for its parabola. a. y = −2( x + 1) 2 + 3 6. c. y = − x 2 + 3 For each of the following equations state the coordinates of the vertex for its parabola. a. y = −2( x + 1) 2 + 3 5. b. y = ( x − 1) 2 + 2 b. y = ( x − 1) 2 + 2 c. y = − x 2 + 3 For each of the following equations complete a table of values for x values from – 2 to 2 and graph. a. y = −( x + 1) 2 + 3 b. y = ( x − 1) 2 + 2 c. y = − x 2 + 3 7. The path of a cliff diver as he dives into a lake, is given by the equation y = −( x − 10) 2 + 75 , where y metres is the diver’s height above the water and, x metres is the horizontal distance travelled by the diver. What is the maximum height the diver is above the water? 8. Graph the exponential function y = 3(2) x where 0 ≤ x ≤ 5 . 9. Complete the table below and graph each function on the same set of axis. Which of the function grows the most rapidly? Which of the functions grows the slowest? x y = 3x y = x3 y = 3x 1 2 3 4 Copyright © 2007, Durham Continuing Education Page 19 of 45 MAP4C Foundations for College Mathematics, Grade 12, College Preparation Lesson 8 Scatter Plots MAP4C – Foundations for College Mathematics Lesson 8 Lesson Eight Concepts ¾ Introduction to scatter plots ¾ Creating scatter plots ¾ Positive, negative and no correlation Scatter plots A scatter plot is a graph of data that is a series of points. Example 1 Study for that Test Here a set of data was plotted. Hours studying vs. Percent of Exam Percent on Exam 100 80 60 40 20 0 0 1 2 3 4 5 6 7 Hours of study Data in a scatter plot can have a positive/negative or no correlation. Negative Correlation No Correlation 120 100 100 80 80 60 60 40 40 20 20 0 0 0 2 4 6 8 6 8 0 2 4 6 8 Positive Correlation 80 60 40 20 0 0 2 4 Copyright © 2007, Durham Continuing Education Page 21 of 45 MAP4C – Foundations for College Mathematics Lesson 8 Example Make a scatter plot for the men’s times. Plot the year on the x-axes and the times on the y-axes. Year 1981 1982 1983 1984 1985 Time 11.82 11.83 11.74 11.64 11.65 Year 1986 1987 1988 1989 1990 Time 11.42 11.51 11.46 11.38 11.36 Year 1991 1992 1993 1994 1995 Time 11.31 11.21 11.09 11.01 10.94 Solution Winning Times for 100m Time (seconds) 12 11.8 11.6 11.4 11.2 11 10.8 1980 1982 1984 1986 1988 1990 1992 1994 1996 Year Support Questions 1. State whether the scatter plot has a positive, negative or no correlation. a. b. Copyright © 2007, Durham Continuing Education c. Page 22 of 45 MAP4C – Foundations for College Mathematics Lesson 8 Support Questions (continued) 2. The table below shows the winning times for the 800-m race at the Olympic Summer Games. Construct a scatter plot. Year 1960 1964 1968 1972 1976 1980 1984 1988 Men’s Time 106.3 105.1 104.3 105.9 103.5 105.4 103 103.45 Key Question #8 1. State whether the scatter plot has a positive, negative or no correlation. b. a. 2. c. The table below shows the winning heights in a men’s high jump competition. Year 1912 1932 1952 1972 1982 2002 Winning Country Canada United States England United States United States Australia Jump in Height (m) 1.93 1.96 1.99 2.16 2.23 2.41 Create a properly labelled Scatter Plot using the data given in the table. Copyright © 2007, Durham Continuing Education Page 23 of 45 MAP4C – Foundations for College Mathematics Lesson 8 Key Question #8 (continued) 3. The table below shows the percent to high school girls who smoked more than one cigarette during the previous year. Graph the data in a scatter plot. Year Percent 1981 34.3 1983 29.5 1985 25.9 1987 25.1 1989 24.4 Copyright © 2007, Durham Continuing Education 1991 22 1993 21.3 1995 21.6 1997 20.4 Page 24 of 45 MAP4C Foundations for College Mathematics, Grade 12, College Preparation Lesson 9 Line of Best Fit and Extrapolation MAP4C – Foundations for College Mathematics Lesson 9 Lesson Nine Concepts ¾ Determining the equation of best fit ¾ Extrapolation ¾ Interpolation Line of Best Fit and Extrapolation Line of Best Fit Line of best fit is a line that passes as close as possible to a set of plotted points. Example Find the line of best fit for the data in the previous example. Solution This is the yintercept. Winning Times for 100m Time (seconds) 12 11.8 11.6 11.4 11.2 11 10.8 1980 1982 1984 1986 1988 1990 1992 1994 1996 Year First pick two points that represent the general center of the points plotted. Then draw the line of best fit through those points generally dividing the plotted point even on both sides. Example What is the approximate equation for the line of best fit in the previous example? Copyright © 2007, Durham Continuing Education Page 26 of 45 MAP4C – Foundations for College Mathematics Lesson 9 Solution Choose the coordinates of the two previously chosen points used to draw the line of best fit. (1985, 11.7) and (1989, 11.4) slope = m = m= y 2 − y 1 11.4 − 11.7 = x 2 − x1 1989 − 1985 − 0. 3 = −0.075 4 y − intercept = b = +12 Therefore the equation of the line of best fit is y = -0.075x +12. Extrapolation and Interpolation Extrapolation is extending a graph to estimate the values that are beyond the table of values. Interpolation is using a graph to estimate the value that are not in the table of values but are within the range of the lowest and largest values presented in the table of values. Example 1 a. Using extrapolation what would be the approximate distance for the discus will be thrown during the 2004 Summer Olympics? b. Using Interpolation how far was the discus thrown in 1952? Distance (m) Women's Olympic Discus Records 80 70 60 50 40 30 20 10 0 1940 1950 1960 1970 1980 1990 2000 2010 2020 Year Copyright © 2007, Durham Continuing Education Page 27 of 45 MAP4C – Foundations for College Mathematics Lesson 9 Solution a. Using extrapolation, what would be the approximate distance for the discus will be thrown during the 2004 Summer Olympics? approximately 78 meters. Women's Olympic Discus Records 80 Draw a vertical line from 2004 until you hit the line of best fit then run horizontally until you hit the value on the y axis Distance (m) 70 60 50 40 30 20 10 0 1940 1950 1960 1970 1980 1990 2000 2010 2020 Year b. Using Interpolation, how far was the discus thrown in 1952? Approximately 51 metres Women's Olympic Discus Records 80 Distance (m) 70 Draw a vertical line from 1952 until you hit the line of best fit then run horizontally until you hit the value on the y axis 60 50 40 30 20 10 0 1940 1950 1960 1970 1980 1990 2000 2010 2020 Year Copyright © 2007, Durham Continuing Education Page 28 of 45 MAP4C – Foundations for College Mathematics Lesson 9 Support Questions 1. The table below shows the winning times for the 100-m Men’s Freestyle Swimming at the Olympic Summer Games. Year 1960 1964 1968 1972 1976 1980 1984 1988 Men’s Time 61.2 59.5 60 58.6 55.7 54.8 55.9 54.9 a. Construct a scatter plot and line of best fit for the data. b. What is the equation of the line of best fit? c. What is the approximate winning time in the year 2020? d. What approximate year does the time drop below 55 seconds? Copyright © 2007, Durham Continuing Education Page 29 of 45 MAP4C – Foundations for College Mathematics Lesson 9 Key Question #9 1. The table below shows Varsity Football Home Game Attendance. Game 1 2 3 4 5 6 7 Attendance 222 285 399 641 529 952 1171 a. Create a properly labelled scatter plot using the data given in the table. b. Draw a line of best fit. c. What is the approximate slope of the line of best fit? d. Using extrapolation, what would the approximate attendance be for game 9? Is this a safe assumption? Explain. 2. The table below shows live bacteria count from a science experiment. Temperature (C) Bacteria (000s) 20 22 24 26 28 30 32 34 36 2.1 4.3 5.2 6.1 6.7 7.6 10.3 8.2 14.1 a. Graph the data in a scatter plot. b. Determine the equation of the line of best fit. c. Use your equation to predict the bacteria count when the temperature reaches 42 °C d. What is the approximate degree, by using extrapolation, to find when the bacteria count when it reaches 18 000? Copyright © 2007, Durham Continuing Education Page 30 of 45 MAP4C Foundations for College Mathematics, Grade 12, College Preparation Lesson 10 Uses and Misuses of Sample Data MAP4C – Foundations for College Mathematics Lesson 10 Lesson Ten Concepts ¾ ¾ ¾ ¾ ¾ Use of scatter plots Use of correlation coefficient Recognizing lack of data Recognizing outliers in a scatter plot Use of extrapolation Uses and Misuses of Sample Data Lack of Data Example 1 Describe the anomaly in the scatter plot given below: Solution The scatter plot above does not have enough data to analyse. There is no minimum that is correct however the more the better. Low correlation/ Low reliability Example 2 Describe the anomaly in the scatter plot given below: No Correlation Copyright © 2007, Durham Continuing Education Page 32 of 45 MAP4C – Foundations for College Mathematics Lesson 10 Solution In this example the correlation value is low, meaning that the points in the scatter plot are all over and not close to the line of best fit. The closer the points to the line of best fit the more accurate the line of best fit is and any conclusions made using the line of best fit. Outliers Example 3 Describe the anomaly in the scatter plot given below: Solution The point in the upper left corner distorts the data and causes the line of best fit to shift slightly towards the outlier causing the distortion. Support Questions 1. Describe the misuse of the line of best fit in each graph below. 2. Give an example of a data set that may be influenced by each factor. a. religion b. society Copyright © 2007, Durham Continuing Education c. sports Page 33 of 45 MAP4C – Foundations for College Mathematics Lesson 10 Support Questions (continued) 3. The number of forest fires in Esexx province is shown below: Year 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Forest Fires in Esexx JuneAug 212 178 287 77 168 318 201 112 322 240 a. Construct a scatter plot and determine whether a trend might exist. b. Is it possible to extrapolate the number of forest fires to expect in the future? Explain. 4. The Ministry of Natural Resources monitors reforestation in Canada. The table below shows the area reforested by seed and seedlings over a six year period; Year 1991 1992 1993 1994 1995 1996 Reforestation (ha) 509 675 465 547 453 701 488 927 459 429 450 173 a. Create a scatter plot and line of best fit on the information given above. b. How strong is the reliability of the line of best fit? Explain. Copyright © 2007, Durham Continuing Education Page 34 of 45 MAP4C – Foundations for College Mathematics Lesson 10 Key Question #10 1. Describe the misuse of the line of best fit in each graph below. 2. Give an example of a data set that may be influenced by each factor. a. politics 3. b. family c. education David is running the 100 m for a high school track and field competition. David’s time trials had him record the time for 9 runs. His times are recorded in the table below. Trial Number 1 2 3 4 5 6 7 8 9 Time (s) 13.2 13.16 12.89 12.91 13.22 15.8 12.93 13.01 12.99 a. Determine a line of best fit for this data b. Are there any outliers in this data, and describe how it/they might have occurred. c. Last year the qualifying time to make the provincial championships was 13.08. Using the model created above do you think David will likely qualify for the championships? 4. The regular season points earned by the President’s Trophy winners over the last 7 years is shown below: Year 99-00 00-01 01-02 02-03 03-04 05-06 06-07 5. points 105 102 118 115 112 108 113 a. Construct a scatter plot and determine whether a trend might exist. b. Draw a line of best fit on your scatter plot. c. Is it possible to extrapolate the number of points to predict the winning points for the next 4 years? Explain. List the 3 anomalies that have been discussed in this lesson that can distort data Copyright © 2007, Durham Continuing Education Page 35 of 45 MAP4C Foundations for College Mathematics, Grade 12, College Preparation Answers to Support Questions MAP4C – Foundations for College Mathematics Support Question Answers Lesson 6: 1. a. 1 b. 25 -3= e. 6 1 1 = 3 216 6 c. f. 2-3 x (16-7)0 = 1 81 1 1 ×1 = 3 8 2 d. 1 g. 7-2= 1 1 = 2 49 7 h. 25 −6 2. ⎛2⎞ ⎛2⎞ a. ⎜ ⎟ × ⎜ ⎟ ⎝5⎠ ⎝5⎠ −3 ⎛ 2⎞ =⎜ ⎟ ⎝5⎠ ( −6)+( −3) c. (m-3)6 = m ( −3 )( 6 ) = m −18 = −2 ⎛ a ( −2 )( −2 ) ⎞ a 4 = ⎜⎜ ( −2 )( −2 ) ⎟⎟ = 4 = 1 ⎝a ⎠ a −5 2 4 −4 ) = (5 ) = 125 3 2 ( 36 ) 3 = (6 ) = 216 3 ( 16 ) 3 = (2) = 8 2 4 3 = 25 2 4 1 ( 8) 4 3 − e. 81 3 4 3 = ( 81) 4 3 4 8 ) = ( 49m 4 n −10 )(m 5 n11 ) = 49m 9 n 1 3 3 = 1 −10 −1 −3 ( c. 16 4 = − 8 3 2 b. 36 = d. 8 4 −2 a. 25 = 1 1 =− 5 4084101 ( −21) 1 m18 (7m n ) (− 3m n ) = (49m n )(−3m n (3mn )(− m n ) − 3m n 2 3. 2−9 59 1953125 357 = −9 = 9 = = 3814 5 2 512 512 = (3ab 2 c )(3ab 2 c ) = 9a 2 b 4 c 2 d. (3ab2c)2 f. −9 = ( −21)( −2 )+( −3 ) = ( −21) −5 = b. (-21)-2 x (-21)-3 ⎛ a −2 ⎞ e. ⎜⎜ −2 ⎟⎟ ⎝a ⎠ ⎛ 2⎞ =⎜ ⎟ ⎝5⎠ 3 1 = (2) 4 = 1 (3) 3 1 16 = 1 27 Copyright © 2007, Durham Continuing Education Page 37 of 45 MAP4C – Foundations for College Mathematics f. 16 − 5 4 = 1 ( 16 ) 5 4 = 1 (2) 5 = Support Question Answers 1 32 1 g. 5 2 + 25 2 = 25 + 25 = 25 + 5 = 30 1 4 1 4 + 5 h. 2 5 × 2 5 = 2 5 4. = 21 = 2 1 6 a. (15625b ) = 6 15625b 6 = 5b 6 1 b. ( −8b 6 c 3 ) 3 = 3 − 8b 6 c 3 = −2b 2 c Lesson 7: 2. a. slope = −4, y- int = 3 2 c. slope = − , y- int = 7 3 a. 3. a. “a” is a positive number (+3) so opens up 1. b. b. slope = 1, y- int = −4 3 d. slope = , y- int = −1 5 c. b. “a” is a negative number (-nothing equates to -1) so opens down c. “a” is a positive number (nothing equates to +1) so opens up d. “a” is a negative number (-nothing equates to -1) so opens down e. “a” is a negative number (-1) so opens down f. “a” is a positive number (+3) so opens up Copyright © 2007, Durham Continuing Education Page 38 of 45 MAP4C – Foundations for College Mathematics Support Question Answers 4. a. (1, 2) b. (3, 0) c. (0, 5) e. (2, 1) 5. a. x = 1 b. x = 3 c. x = 0 6. a. d. (0, 0) Copyright © 2007, Durham Continuing Education f. (4, -2) Page 39 of 45 MAP4C – Foundations for College Mathematics 6. Support Question Answers b. Copyright © 2007, Durham Continuing Education Page 40 of 45 MAP4C – Foundations for College Mathematics 6, c. 7. b and e (both represent a rapid increase or decrease) Support Question Answers 8. x 0 1 2 3 4 5 y 2 6 18 54 162 486 Copyright © 2007, Durham Continuing Education Page 41 of 45 MAP4C – Foundations for College Mathematics Support Question Answers Lesson 8: 1. a. no correlation b. positive correlation c. negative correlation 2. Lesson 9: 1. a. Copyright © 2007, Durham Continuing Education Page 42 of 45 MAP4C – Foundations for College Mathematics Support Question Answers b. (1964,59.5) and (1984,55.9) slope = m = m= y 2 − y 1 55.9 − 59.5 = x 2 − x 1 1984 − 1964 − 3 .6 = −0.18 20 y − intercept = b = +61 Therefore the equation of the line of best fit is y = -0.18x +61. c. ≈ 49 .5s Copyright © 2007, Durham Continuing Education Page 43 of 45 MAP4C – Foundations for College Mathematics d. Support Question Answers about 1990 Lesson 10: 1. a. No or little correlation. Points are too spread out and line of best fit has little reliability. b. The point in the upper left corner is an outlier and distorts the date and the line of best fit. c. No enough data to support the reliability of the line of best fit. 2. a. Answers vary: A survey asking whether stores should be open on Sundays. b. Answers vary: The amount of recycled materials sent to recycling. c. Answers vary: A survey asking what individuals like to do with their spare time. Copyright © 2007, Durham Continuing Education Page 44 of 45 MAP4C – Foundations for College Mathematics 3. Support Question Answers a. appears to be no trend existing b. Since there is no trend it would be highly inaccurate when making any conclusions based on extrapolating any line of best fit. 4. a. b. Weak reliability: few data points and what data that is present is fairly spread out. Copyright © 2007, Durham Continuing Education Page 45 of 45