Step 1. Use a ruler or straight-edge to determine a line of best fit. One example is shown below.

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1 Linear Models Modeling 1 ESSENTIALS Example Draw a straight line through the scatter plot so that the line represents a best fit approximation to the points. Then determine the equation for the line drawn. Note. The straight line does not need to go through every point but it should be close to most of the points. Step 1. Use a ruler or straight-edge to determine a line of best fit. One example is shown below. Step. Give the equation for the line. Use slope-intercept form: y = mx + b b = 50 since the y-intercept is (0, 50). Select two points on the line and use them to determine the slope. The points, (6, 103) and (100, 45), produce a slope of m = = = The equation for the line of best fit is approximately y = 1.9x If you used different points, you may have come up with a different equation. However, the values should be close to the calculated equation shown above.

2 Linear Models Modeling GUIDED LEARNING EXAMPLE 1 Draw a straight line through the scatter plot so that the line represents a best fit approximation to the points. Then determine the equation for the line drawn. Step 1. Use a ruler or straight-edge to determine a line of best fit. Draw the line onto the graph shown above. Step. Give the equation for the line. Use slope-intercept form: y = mx + b b = since the y-intercept is (0, ). Select two points on the line and use them to determine the slope. Point 1: (x1, y1) = (, ) and Point : (x, y) = (, ) Calculate the slope of the line between the points: m = y y 1 x x 1 = = The equation for the line of best fit is approximately y = x +.

3 Quadratic Models Modeling 3 ESSENTIALS Example Draw a -shaped curve through the scatter plot so that the line represents a best fit approximation to the points. Then determine the equation for the line drawn. Note. The -shaped curve does not need to go through every point but it should be close to most of the points. Step 1. The vertex of a quadratic model is the highest (or lowest) point on the graph. The graph of the quadratic model should look like mirror images on each side of an imaginary vertical line which runs through the vertex. Notice that the vertex of the quadratic model, shown below, is at the point (0, -5). (0, -5) (1, -8) Step. The -shaped curve is formed by squaring the x- value. All of our quadratic models will fit the equation y = ax + k, where a and k are real numbers. In this step, we need to identify at least two points on the graph. Be sure to use the vertex and then use at least one additional point. The vertex is at the point (0, -5). The point, (1, -8) is also on the graph. Step 3. Determine the values for a and k that make the equation y = ax + k true. Substitute (0, -5) into the equation and solve for k: 5 = a(0) + k simplifies to 5 = k Substitute (1, -8) into the equation, y = ax 5, and solve for a: 8 = a(1) 5 8 = a = a 3 = a The quadratic model that fits the graph is y = 3x 5.

4 Quadratic Models Modeling 4 GUIDED LEARNING EXAMPLE 1 ESSENTIALS Example Draw a -shaped curve through the scatter plot so that the line represents a best fit approximation to the points. Then determine the equation for the line drawn. Note. The -shaped curve does not need to go through every point but it should be close to most of the points. Step 1. The vertex of a quadratic model is the highest (or lowest) point on the graph. The graph of the quadratic model should look like mirror images on each side of an imaginary vertical line which runs through the vertex. Our models will have a vertex with an x-value equal to zero. Draw the -shaped curve with an x-value of zero that is a best fit for the data plotted in the graph shown above. Step. Identify the location of the vertex and at least one additional point for the quadratic model drawn on the graph above. The vertex is at the point (0, ). The point, (, ) is also on the graph. Step 3. Determine the values for a and k that make the equation y = ax + k true. Substitute the vertex into the equation and solve for k: = a(0) + k simplifies to = k Substitute the other point into the equation, y = ax +, and solve for a: The quadratic model that fits the graph is y = x +.

5 Exponential Models Modeling 5 ESSENTIALS Exponential Models. There are two basic types of exponential models: exponential growth and exponential decay. In an exponential model, the value for y increases (or decreases) proportionally to its current value. That means that to calculate the next y-value, you multiply the previous y-value by a constant. In an exponential growth model, the y-values are multiplied by a value larger than one in order to get the next y-value. In an exponential decay model, each y-value is multiplied by a value between zero and one. Exponential Growth Exponential Decay y = a b x, b > 1 y = a b x, 0 < b < 1 Example Use the scatter plot shown to the right to determine the exponential equation of the form, y = a b x, of best fit. Step 1. Determine whether this is an exponential growth model or an exponential decay model. The graph is an exponential growth model since the graph is increasing, or rising to the right. Step. Draw the exponential line of best fit. Then plot and label the y-intercept and at least one additional point on the graph of the line. Step 3. Determine the values for a and b that make the equation y = a b x true. Substitute (0, ) into the equation and solve for a: = a b 0 simplifies to = a Substitute (, 8) into the equation, y = b x, and solve for b: 8 = b Divide each side of the equation by : 4 = b. Thus, b =. (0, ) (, 8) The exponential model that fits the graph is y = x.

6 Exponential Models Modeling 6 GUIDED LEARNING EXAMPLE 1 ESSENTIALS Example Use the scatter plot shown to the right to determine the exponential equation of the form, y = a b x, of best fit. Step 1. Determine whether this is an exponential growth model or an exponential decay model. The graph is an exponential model since the graph is decay or growth, or to the right. decreasing or increasing falling or rising Step. Draw the exponential line of best fit. Then plot and label the y-intercept and at least one additional point on the graph of the line. The y-intercept is at the point (0, ). The point, (, ) is also on the graph. Hint: For the second point, select a point with an x-value of either 1 or -1. This will give you an easier equation to solve in Step 3. Step 3. Determine the values for a and b that make the equation y = a b x true. Substitute (0, ) into the equation and solve for a: = a b 0 simplifies to a = Substitute the other point into the equation, y = b x, and solve for b: The exponential model that fits the graph is y =. x

7 Exponential Equations Matching Activity Modeling 7 Cut out each square. Then match each graph, table, and equation. y y 4 x a f k b c d e g h i j 1 y x y x x y x y x l m n o

8 Quadratic Equations Matching Activity Modeling 8 Cut out each square. Then match each graph, table, and equation y y x x y( x ) y ( x ) y x

9 Linear Equations Matching Activity Modeling 9 Cut out each square. Then match each graph, table, and equation. A B C D E F G H I J y 4x y x 4 y 4x y 4x y x 4 N K L M O

10 Mixed Equations Matching Activity Modeling 10 This activity should be paired with the squares from the previous three pages. Match as many squares as possible to each graph, table, and/or equation. Here is one discussion question but there are many others that you can bring up: Some categories are mutually exclusive where one graph, table, or equation can belong to just one category but not the other. Other categories are more generalizes and one graph (or table or equation) can belong to both categories. Which categories are mutually exclusive and which are not? Why? The graph is always decreasing The graph increases for some time and then begins decreasing The graph is a straight line I V IX The graph is always increasing The graph is U-shaped The graph is exponential II VI X The y-intercept is negative The y-intercept is positive The x-intercept is positive III IV The x-intercept is negative VII Sam earns $4 each hour. Which graph, table, and equation illustrates how much Sam will earn during an 8-hour day? VIII XI Sallie has a magic ball that bounces feet on its first bounce, 4 feet on the second bounce, 8 feet on the third bounce, and so on. Which graph, table, and equation illustrates how high Sallie s ball bounces for XII the first 8 bounces?

11 Payroll Nightmare Modeling 11 Genaro, an employee of Bouncy Balls, Inc., makes the following proposition to the company regarding his salary. Should the company accept his proposition? Genaro, a new employee, is offered a weekly salary of $000. Instead of accepting the weekly salary, Genaro makes the following counteroffer: Pay me just one penny the first week, two pennies the second week, four pennies the next week, and so on, doubling my salary each week. Which would be the better deal for Genaro and which would be the better deal for the company? Let s explore by making a Table of Values. 1) Fill in the Table of Values for the different salary options. Using the first salary option, Genaro will receive $000 each week. His weekly salary does not change. Find the cumulative salary by adding the $000 earned each week. In option, Genaro s salary doubles each week. Complete the table for the first 10 weeks. Use the last row to determine the formulas for the first 3 columns. You may want to use a calculator and the formula. Company Offer Genaro s Counter Offer Week # Salary Option 1: $000/wk (dollars) Total Cumulative Salary Company Offer (dollars) Salary Option : Doubling (cents) Total Cumulative Salary Counter Offer (cents) x x-1 x 1

12 Payroll Nightmare Modeling 1 ) Graph the cumulative weekly salary for both options on each of the graphs given below. For consistency, enter the values for Option using dollars instead of cents. Option 1: $000 each week Option : Doubling 3) Determine the equation which gives the cumulative weekly salary for Week #x for both salary options. Label each graph type as either linear or exponential. Option 1: Equation: Type: Option : Equation: y = 0. 01( x 1) Type: 4) Which salary option is the best choice for the company? Why?

13 Payroll Nightmare Modeling 13 5) Use the two equations, from question #3, to complete the following tables of values. Week # 10 Total Cumulative Salary Company Offer (dollars) Total Cumulative Salary Counter Offer (dollars) ) Approximately how many weeks would Genaro have to work before Option # becomes the best salary option for him?

14 Challenging. Payroll Nightmare. Modeling 14 The graph shown below represents Celeste s salary over the past ten years. Each year, the same formula has been used to calculate her salary. Find the formula and use it to determine Celeste s starting salary and her salary for the upcoming year. Graphical Tabular x y y = Annual Salary x = number of years since initial hire Hint Solution Celeste s annual salary for her 11 th year will be $. Calculation The graph appears to be exponential. A basic exponential graph has an equation of the form Use this space to work out your answer. y = a b x Use the graph to find the y-intercept. Substitute the y-intercept into the equation and solve for a. Then use a nd point to determine the value for b.

15 Profit and Loss Modeling 15 Profit = Revenue Costs Profits are obtained when the costs are lower than the revenue. Losses are obtained when the costs are higher than the revenue. A loss occurs when the profit is negative. Terry, Bouncy Ball, Inc. s account manager, is concerned about company profits and losses. George, the production supervisor, and Julie, the sales manager, have provided the following cost and revenue equations, respectively: Cost: x Revenue: 150x In each case, x represents x thousand Bouncy Balls. 1. Use the cost and revenue equations to determine the profit equation. Profit = Revenue Costs. Use the profit equation in order to calculate the amount of profit for each level of production. Number of Balls Produced and Sold Total Profit/Loss x 1 3. A positive profit indicates that the company is making money. A negative profit indicates that the company is losing money. At what point of production, does the company begin making money? 4. The break even point is the point at which profits are equal to zero. Find the break even point. 5. Does the profit equation have a positive or a negative slope? How does this impact profits as the production of the number of Bouncy Balls increases?

16 Height of the ball (in feet) Path of a Bouncy Ball Modeling 16 A Bouncy Ball is thrown vertically upward from the ground. After seconds the ball attains its maximum height of 10 feet. After 4 seconds it hits the ground. 1) Draw a rough sketch of the path of the Bouncy Ball over time by filling in the Table of Values shown below and then plotting the points on the graph using a smooth shape curve. Be sure to plot and label each point on the graph. Time after the Ball is thrown Height of the Ball 0 seconds seconds 4 seconds ) Estimate the location for the vertex of the parabola for the equation shown in the graph. Note: The vertex is the turning point. Vertex: (h, k) = _(, ) Time of the ball in the air (in seconds) 3) Use the graph that you created in order to estimate the height of the ball at each of the following times. Plot each point on your graph. Time after the Ball is thrown Height of the Ball ½ second 1 second 1 ½ seconds ½ seconds 3 seconds 3 ½ seconds

17 Path of a Bouncy Ball Modeling 17 4) Let t represent the time of the ball in the air and let y represent the height of the ball at time t. Then the equation that represents the path of the ball is y = 3(t h) + k, where the location of the vertex is (h, k). a) Give the coordinates for the vertex of the graph drawn in question #: (h, k) = b) Now use the coordinates of the vertex to complete the formula that models the height of the ball. h k y = 3(t ) + 5) Use the equation from #4 to determine the actual height of the ball at each of the following times. Time after the Ball is thrown Height of the Ball 0 seconds ½ second 1 second 1 ½ seconds seconds ½ seconds 3 seconds 3 ½ seconds 4 seconds 6 seconds 6) How does your graph compare to your model?

18 Exercise Set Math Models Modeling 18 Refer to graphs a f to answer each question. a. b. c. d. e. f. 1. Which graphs have a line of best fit that is linear?. Which graphs have a line of best fit that is quadratic? 3. Which graphs have a line of best fit that is exponential? 4. Which graphs have a line of best fit that is always increasing? 5. Which graphs have a line of best fit that is always decreasing? 6. Which graphs have a line of best fit that is sometimes increasing and sometimes decreasing? 7. Which graphs have a line of best fit with a positive y-intercept? 8. Which graphs have a line of best fit with a negative y-intercept? 9. Use the graphs to match an equation of best fit with one of the equations, A-F, shown below. A. y = x 6 C. y = x + 3 E. y = x B. y = 4 ( 1 4 )x D. y = 4x F. y = 3x + 4

19 Answer Key Modeling 19 Linear Models Answers may vary. Example 1. b = 7, y-intercept: (0, 7) Points chosen: (x1, y1) = (4, -1) and Point : (x, y) = (8, -3) Quadratic Models Answers may vary. m = y y 1 = 3 ( 1) = 0 = 5 x x The equation for the line of best fit is y = 5x + 7. Example 1. The vertex is at the point (0,-11). The point, (4, -) is also on the graph. 11 = a(0) + k; 11 = k Exponential Models Answers may vary. = a(4) 11 ; a = 9/16. The quadratic model that fits the graph is y = 9 16 x 11. Example 1. The graph is an exponential decay model since the graph is decreasing or falling to the right. The y-intercept is at the point (0, 5). The point, (-1, 10) is also on the graph. Substitute (0, 5) into the equation and solve for a: 5 = a b 0 simplifies to a = 5. Substitute the other point into the equation, y = 5 b x, and solve for b: The exponential model that fits the graph is y = 5 ( 1 )x Exponential Equations Matching Activity a, g, o; b, f, m; c, I, n; d, h, k; e, j, l Quadratic Equations Matching Activity 1, 7, 1;, 10, 15; 3, 6, 14; 4, 9, 11; 5, 8, 13 Linear Equations Matching Activity A, H, M; B, I, N; C, G, L; D, J, O; E, F, K

20 Answer Key Modeling 0 Mixed Equations Matching Activity Payroll Nightmare I: A, D, H, J, M, O, a, c, d, g, h, I, k, n, o; II: B, C, E, F, G, I, K, L, N, b, e, f, j, l, m; III: A, B, D, H, I, J, M, N, O, a, b, c, e, f, g, i, j, k, m, n, o; IV: B, I, N; V: 1,, 4, 7, 9, 10, 11, 1, 15; VI: 1,, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 13, 14, 15; VII: C, G, L, d, h, k,, 10, 15; VIII: E, F, K; IX: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O; X: a, b, c, d, e, f, g, h, I, j, k, l, m, n, o; XI: A, C, D, G, H, J, L, M, O, 4, 9, 11; XII: e, j, l 1) 4: 000, 8000, 8, 15 5: 000, 10,000, 16, 31 6: 000, 1,000, 3, 63 7: 000, 14,000, 64, 17 8: 000, 16,000, 18, 55 9: 000, 18,000, 56, : 000, 0,000, 51, 103 x: 000, 000x, x-1, x 1 1) ) Option 1: Equation: y = 000x; Type: Linear Option : Equation: y = 0.01( x 1); Type: Exponential 3) Answers Vary. Possible Answers: Linear is best since the salary is consistent. After weeks, the linear salary is cheaper for the company. 4) 10: $0,000, $ : $30,000, $ : $40,000, $10,486 5: $50,000, $335,544 30: $60,000, $10,737, : $80,000, $10,995,116, : $100,000, $ ) His 3 rd week.

21 Height of the ball (in feet) Answer Key Modeling 1 Payroll Nightmare. Answers may vary; The answer should be anywhere from $38,180 to $59,1. Profit and Loss. 1) Profit = 148.8x 9000 ) Number of Balls Produced and Sold x Total Profit/Loss ) x = 8 (producing and selling 8000 balls) 4) x = ) Positive; Profit increases as production increases. Path of a Bouncy Ball. 1) (, 10) Time after the Ball is thrown Height of the Ball 0 seconds 0 seconds 10 4 seconds 0 (0, 0) (4, 0) ) Vertex: (h, k) = (, 10) 3) Time of the ball in the air (in seconds) Time after the Ball is thrown Height of the Ball ½ second 60 1 second ½ seconds 115 ½ seconds 110 The answers may vary. 3 seconds ½ seconds 60

22 Answer Key Modeling 4) 4a) (h, k) = (, 10) 4b) y = 3(t ) ) Time after the Ball is thrown Height of the Ball Exercise Set 0 seconds -8 ½ second 48 1 second 88 1 ½ seconds 11 seconds 10 ½ seconds 11 3 seconds 88 3 ½ seconds 48 4 seconds -8 6 seconds -39 6) The graph will be more accurate close to the vertex and less accurate at the outside branches. 1) a, c ) e, f 3) b, d 4) c, d 5) a, b 6) e, f 7) a, b, d, f 8) c, e 9) A: e; B: b; C: f; D: c; E: d; F: a