Up and Down or Down and Up

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1 Lesson.1 Assignment Name Date Up and Down or Down and Up Exploring Quadratic Functions 1. The citizens of Herrington County are wild about their dogs. They have an existing dog park for dogs to play, but have decided to build another one so that one park will be for small dogs and the other will be for large dogs. The plan is to build a rectangular fenced in area that will be adjacent to the existing dog park. The sketch is shown below. The county has enough money in the budget to buy 1000 feet of fencing. w Existing Dog Park New Dog Park l w a. Determine the length of the new dog park, l, in terms of the width, w. 2w 1 l l w b. Write a function for the area of the new dog park, A(w), in terms of the width, w. Write the function in standard quadratic form. Does this function have an absolute minimum or an absolute maximum? Explain your answer. A(w) 5 l? w A(w) 5 ( w) w A(w) w 2 2 w 2 A(w) 5 22 w w This function has an absolute maximum. Because the coefficient of the squared-term is negative, the parabola opens downward which results in an absolute maximum. Chapter Assignments 153

2 Lesson.1 Assignment page 2 c. Determine the x-intercepts of the function. Explain what each means in terms of the problem situation. The x-intercepts are (0, 0) and (500, 0). Each represents the width in feet for which the area of the rectangular park is 0 square feet. The x-intercept (0, 0) means that if the width of the park is 0 feet then the area will be 0 square feet, which makes sense because there will be no enclosed area if there is no width. The x-intercept (500, 0) means that if the width of the park is 500 feet then the area will be 0 square feet. This makes sense because if both sides of the park are 500 feet wide, then there will be no fencing left for the length so there will be no area to enclose. d. What should the dimensions of the dog park be to maximize the area? What is the maximum area of the park? The width of the dog park should be 250 feet and the length of the dog park should be (250) feet. The maximum area of the park is 125,000 square feet. e. Sketch the graph of the function. Label the axes, the absolute maximum or minimum, the x-intercepts, and the y-intercept. Area (square feet) y 180, , , , ,000 80,000 60,000 40,000 20,000 (0, 0) (250, 125,000) (0, 500) Width (feet) f. Use the graph to determine the dimensions of the park if the area was restricted to 105,000 square feet. If the area is 105,000 square feet the width could either be 150 feet or 350 feet. So the park could either be 150 feet wide and (150) feet long or the park could be 350 feet wide and (350) feet long. x 154 Chapter Assignments

3 Lesson.2 Assignment Name Date Just U and I Comparing Linear and Quadratic Functions 1. The Quickgrow Fertilizer Company is working on different formulas for flower fertilizers. The table shows the growth of unfertilized plant A and the growth of fertilized plant B. Time (days) Height of plant A (centimeters) Height of plant B (centimeters) a. Which plant height would be represented by a linear function? Which would be represented by a quadratic function? Explain your reasoning. The height of plant A would be represented by a linear function because the first differences are all the same and the second differences are all 0. The height of plant B would be represented by a quadratic function because the first differences are changing but the second differences are all the same. b. Would the function A(x) 5 22x 1 4 or A(x) 5 2x 1 4 represent the growth of plant A? Explain using leading coefficients. The linear function A(x) 5 2x 1 4 would represent the growth of plant A because the height of the plant is increasing as the number of days increases so the leading coefficient must be positive. Chapter Assignments 155

4 Lesson.2 Assignment page 2 c. Would Graph A or Graph B represent the growth of plant B? Explain using second differences. y Height (centimeters) 36 Graph B Graph A Number of Days x Graph B would represent the growth of Plant B. Because the second differences in the plant growth are positive, the graph would open upward. 2. The Quickgrow Fertilizer Company has run into problems while experimenting with a type of fertilizer that is supposed to increase yield of pepper plants. The yield for plant C can be represented by the function C(x) x The yield for plant D can be represented by the function D(x) 5 23x x The graphs of the yields for both plants are shown. y Yield (number of peppers) Fertilizer (tsp) x 156 Chapter Assignments

5 Lesson.2 Assignment page 3 Name Date a. Determine the y-intercept(s) of each function and describe the meaning of each in terms of the problem situation. The y-intercept for the linear function is at (0, 100). This means that when no fertilizer was put on the plant it yielded 100 peppers. The y-intercept for the quadratic function is at (0, 50). This means that when no fertilizer was put on the plant it yielded 50 peppers. b. Determine the x-intercept of the linear function algebraically. Describe the meaning in terms of the yield for plant C. C(x) x x x 8 5 x The x-intercept of the linear function is at (8, 0). This means that when 8 teaspoons of the fertilizer were put on plant C it did not yield any peppers. c. Determine the x-intercept(s) of the quadratic function. Then, describe the meaning of each in terms of the problem situation. The x-intercepts of the quadratic function are at about (21.9, 0) and (8.9, 0). The first intercept does not make sense because you cannot put a negative amount of fertilizer on the plant. The second intercept means that when 8.9 teaspoons were put on the plant it did not yield any peppers. d. Determine the absolute maximum of the quadratic function. Explain what it means in terms of the yield for plant D. The absolute maximum of the quadratic function is at (3.5, 86.75). This means that plant D would yield a maximum of about 87 peppers if 3.5 teaspoons of fertilizer were put on it. Chapter Assignments 157

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7 Lesson.3 Assignment Name Date Walking the... Curve? Domain, Range, Zeros, and Intercepts 1. A masking tape company has to decide how many hundreds of rolls of tape to produce each day. The company knows that the costs to produce the tape go down the more rolls they make. However, the overall cost to the company increases if they make too many rolls due to the cost of storing overstock. The company determined that the cost to produce x hundreds of units a day could be represented by the function f(x) x x 1 15,000. a. Graph the function. Sketch the graph and label the axes. y Cost (dollars) 45,000 40,000 35,000 30,000 25,000 20,000 15,000 10, Rolls of Tape x b. What are the domain and range of the function in terms of the graph? The domain of the function is all real numbers from negative infinity to positive infinity. The range of the function is all real numbers greater than or equal to 13,400. c. What are the domain and range of the function in terms of the problem situation? The domain in terms of the number of rolls of tape produced is any integer greater than or equal to 0. The company cannot make a negative number of rolls of tape or parts of rolls of tape. The range in terms of the profit is all real numbers greater than or equal to 13,400. This means that the costs associated with making rolls of tape will never be less than $13,400. Chapter Assignments 159

8 Lesson.3 Assignment page 2 d. Over what interval does the cost of making the rolls of tape decrease? Increase? The cost of making the rolls of tape decreases over the interval [0, 200) and increases over the interval (200, `). e. How many rolls of tape should the company make to minimize the cost? The company should make 20,000 rolls of tape daily to minimize cost. f. What is the minimum cost to the company? What does this number represent for the function? The minimum cost is $13,400. This is the absolute minimum of the function. g. Determine the x-intercept(s) of this function and describe what they mean in terms of the cost to the company. There are no x-intercepts for this function. This means that the cost of producing the rolls of tape will never equal 0. This is because of fixed costs associated with overhead, wages, etc. 2. The profit a masking tape company makes from producing and selling x hundred rolls of tape can be represented by the function g(x) x x 2 15,000. The graph of the profit function is shown. y 40,000 30,000 20,000 10, , , , , x a. What is the domain of this function? What is the domain for the problem situation? The domain of the function is all real numbers. The domain for the problem situation is all integers greater than or equal to zero. They cannot make a negative number of rolls of tape or parts of rolls. 160 Chapter Assignments

9 Lesson.3 Assignment page 3 Name Date b. What is the range of this function? What is the range for the problem situation? The range of the function is all real numbers less than or equal to 10,000. The range for the problem situation is all real numbers less than or equal to 10,000. c. Over what interval does the profit increase? Decrease? The profit increases over the interval [0, 500). The profit decreases over the interval (500, `). d. How many rolls of tape must they produce and sell to make a profit of $1590? They must make either 21,000 or 79,000 rolls of tape to make a profit of $1590. e. Determine the x-intercepts of the function. Describe what the x-intercepts mean in terms of this problem situation. The x-intercepts are about (183.77, 0) and (816.23, 0). The x-intercepts indicate that if the company makes around 18,377 or 81,623 rolls of tape they will break-even, or have zero profit. f. Over what interval(s) is there a negative profit? Over what interval(s) is there a positive profit? The company has a negative profit over the intervals [0, ) and (816.23, `) and a positive profit over the interval (183.77, ). Chapter Assignments 161

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11 Lesson.4 Assignment Name Date Are You Afraid of Ghosts? Factored Form of a Quadratic Function 1. The Squeaky Clean Car Wash charges $20 for its deluxe wash. At this price, the car wash averages 200 customers per day. The car wash has determined that for every $0.25 that they decrease the price of the wash, they will see an increase of 5 customers per day. a. Complete the table to determine the revenue the car wash will see based on the number of price decreases. Number of Price Decreases Price of Car Wash (dollars) Number of Customers Revenue (dollars) n n n ( n)( n) b. The manager of the car wash thinks that he should just keep decreasing the price to bring in the most customers. Based on the table, is this a good decision? Explain your reasoning. No. The manager should not keep decreasing the price. At some point the price will be too low and even though the numbers of customers increase the revenue starts to decrease. c. Write the revenue for the car wash as a function, R(n), in standard form. Then, identify a, b, and c for the function. R(n) 5 ( n)( n) R(n) 5 20(200) 1 20(5n) 1 (20.25n)(200) 1 (20.25n)(5n) R(n) n 2 50n n 2 R(n) n n a , b 5 50, and c Chapter Assignments 163

12 Lesson.4 Assignment page 2 d. Based on the equation, will the graph of the revenue function open up or down? Explain your reasoning. The graph of the revenue will open down because the coefficient of the x 2 term is negative. e. Graph the revenue function. Then sketch the graph and label the axes. y Revenue (dollars) Number of Decreases f. At what number of price decreases will the revenue be maximized? What is the maximum revenue the car wash will see at that price? The revenue will be maximized when there are 20 price decreases. The maximum revenue will be $4500. x g. What should the price be to maximize revenue? How many customers will the car wash see at that price? Show your work (20) The price should be $15 to maximize revenue (20) The car wash will see 300 customers when the price is $15. h. Determine the x-intercepts of the revenue function. Explain what they mean in terms of the problem situation. The x-intercepts are 240 and 80. The intercept 240 does not make sense to this problem because there cannot be a negative number of decreases in price. The intercept 200 means that after 200 price decreases, the revenue will equal $0. This is because the price of the car wash will be $0 after 200 price decreases. i. Determine the y-intercept of the revenue function. Explain what it means in terms of the problem situation. The y-intercept is This means that if there are 0 price decreases, the car wash can expect revenue of $ Chapter Assignments

13 Lesson.5 Assignment Name Date Just Watch that Pumpkin Fly! Investigating the Vertex of a Quadratic Function Investing in the stock market is always a risk. Sometimes there can be big payouts but other times you can end up losing it all. 1. Maya has saved up some money and decides to take a risk and invest in some stocks. She invests her money in Doogle, a popular computer company. Unfortunately she lost it all over a matter of months. The change in her money during this investment can be represented by the function v(x) x 2 3 x 2, where v is the value of her investment and x is the time in months. a. How much money did Maya first invest in the company? What does this value represent in the function? Maya starts with $75. The value (0, 75) represents the y-intercept of the function. b. Determine the x-intercepts of the function. Explain what each intercept means in terms of the problem situation. The x-intercepts are (21, 0) and (25, 0). The x-intercept (21, 0) means that she had 0 dollars one month before investing in the company. There is no way to know that, so this does not make sense. The x-intercept (25, 0) means that after 25 months the value of her investment was 0 dollars. This makes sense to the problem situation because Maya lost all her money. c. Determine the vertex. Explain what it means in terms of the problem situation. The vertex is (12, 507). This means that her investment reached its maximum value of $507 after 12 months. d. Determine when her portfolio reached a value of $360. The portfolio reached a value of $360 when it had been invested for 5 months and when it had been invested for 19 months. Chapter Assignments 165

14 Lesson.5 Assignment page 2 2. Jack invested some of his money in Home-mart, a large home improvement store in his town. A few years after investing, the company went out of business and Jack lost all his money. The growth and decline of his money over this time can be represented by the function v(x) 5 22 x x 1 100, where v is the value of his investment and x is the time in months. a. Describe this function in terms of the problem situation. Include information regarding the y-intercept, the x-intercepts, and whether the function has an absolute maximum or absolute minimum. Jack starts with $100. The value (0, 100) represents the y-intercept of the function. The x-intercepts are (21, 0) and (50, 0). The x-intercept (21, 0) means that he had 0 dollars one month before investing in the company. There is no way to know that, so this does not make sense. The x-intercept (50, 0) means that after 50 months the value of his investment was 0 dollars. This makes sense in the problem because Jack ended up losing all his money. The function has an absolute maximum. This represents when the investment was at its greatest. b. Determine the axis of symmetry for this parabola. Then determine the vertex and explain what it means in terms of the problem situation. The axis of symmetry is x because The y-coordinate when x is: v(24.5) 5 22(24.5 ) (24.5) The vertex is (24.5, ). This means that when the money has been in the portfolio for 24.5 months the value will be a maximum at $ Chapter Assignments

15 Lesson.5 Assignment page 3 Name Date c. Jack s account has $1288 in it after 22 months. Use the axis of symmetry to determine another time when the account will have $1288 in it. Show and explain your work. Another point on the parabola with a y-value of $1288 is a symmetric point to (22, 1288). The x-coordinate is: 22 1 a a 5 49 a 5 27 The account will also have $1288 in it when the money has been in the account for 27 months. d. Draw a graph of this function on the grid provided. Label the axes, vertex, axis of symmetry, x-intercepts, and the set of symmetric points that you determined. Value of Portfolio (dollars) (22, 1288) y (24.5, ) (27, 1288) (50, 0) Time (months) 200 (21, 0) x x Chapter Assignments 167

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17 Lesson.6 Assignment Name Date The Form Is Key Vertex Form of a Quadratic Function 1. A concert venue can hold up to 20,000 people. The concert will sell out if tickets are sold for $40 a piece. In order to make more money, the venue would like to increase the ticket price. They determine that for every one dollar increase in price, 200 fewer people will attend. If x represents the number of one dollar increases in the price, then the revenue that the concert will bring in is represented by the function R(x) 5 (20, x)(40 1 x). a. Rewrite this function in the correct factored form. Then state the key characteristics you can determine from the equation and what they mean in terms of the problem situation. R(x) (x 2 100)(x 1 40) The parabola opens downward, which means that the revenue will increase, reach a maximum, and then decrease again as the number of one dollar increases in price goes up. The x-intercepts are (100, 0) and (240, 0). The first intercept (100, 0) means that if there are 40 one dollar price increases, then the revenue will be 0 dollars because no one will come to the concert. The second intercept (240, 0) does not make sense because there cannot be a negative number of one dollar price increases. b. Determine the vertex for the function. Explain what it means in terms of the problem situation. Then use it to rewrite the function in vertex form. The vertex is at (30, 980,000). This means when there are 30 one dollar increases in price, the revenue will be at a maximum of $980,000. The vertex form is R(x) (x 2 30 ) ,000. c. Determine the y-intercept for the function. Explain what it means in terms of the problem situation. The y-intercept is (0, 800,000). This means that when there are not any one dollar increases in price the revenue will be $800,000. d. Which of the following functions must be the revenue function written in standard form? Explain your reasoning. R(x) x ,000x 2 800,000 or R(x) x ,000x 1 800,000 The revenue function written in standard form must be the second function, R(x) x ,000x 1 800,000, because this function has a positive y-intercept, meaning when there are 0 one dollar increases the revenue is a positive amount. Chapter Assignments 169

18 Lesson.6 Assignment page 2 2. Perez throws a softball up in the air. The height of the ball in meters can be determined by the function h(t) (t 2 3 ) , where t is the time it is in the air in seconds. a. Identify the form of this quadratic function. Then state all you know about the key characteristics, based only on the given equation of the function. Explain what they mean in terms of the problem situation. This quadratic function is in vertex form. The parabola opens downward, which means that the height of the ball will increase, reach a maximum, and then decrease again as time increases. The vertex of the parabola is (3, 60). This means that when the ball has been in the air 3 seconds, it will reach a maximum height of 60 meters. b. Determine the x-intercept(s) of the function. Explain what they mean in terms of the problem situation. Then, write the function in factored form. The x-intercepts are about (20.5, 0) and (6.5, 0). The first intercept does not make sense to the problem because time cannot be negative. The second intercept means that the ball will hit the ground in 6.5 seconds. The function in factored form is h(t) (t 1 0.5)(t 2 6.5). c. Use your graphing calculator to determine the y-intercept for the function. Explain what it means in terms of the problem situation. The y-intercept of the function is about (0, 15.9). This means that the initial height of the ball was 15.9 meters. d. Which of the following functions must be the revenue function written in standard form? Explain your reasoning. h(t) t t or h(t) t t The height function written in standard form must be the first function, h(t) t t , because this function has a positive y-intercept, meaning when time is zero the height of the ball is 15.9 meters. 170 Chapter Assignments

19 Lesson.6 Assignment page 3 Name Date 3. A company knows that the more it advertises the more product it will sell. However, advertising more will also cost more money, which then takes away some of the profit. The profit will therefore follow the path of a parabola, because it will increase from more advertising but eventually decrease if too much money is spent on advertising. The profit (in thousands of dollars) can be represented by the function P(x) 5 22 x x 1 60, where x represents the amount of money spent (in thousands of dollars). a. Identify the form of this quadratic function. Then state all you know about the key characteristics, based only on the given equation of the function. Explain what they mean in terms of the problem situation. This quadratic function is in standard form. The parabola opens downward, which means that the profit will increase, reach a maximum, and then decrease again as the amount of money spent on advertising increases. The y-intercept of the parabola is (0, 60). This means that when the company spends 0 dollars on advertising the profit will be $60,000. b. Determine the x-intercept(s) of the function. Explain what they mean in terms of the problem situation. Then, write the function in factored form. The x-intercepts are (23, 0) and (10, 0). The first intercept does not make sense to the problem because money cannot be negative. The second intercept means that when $10,000 is spent on advertising the profit will be 0 dollars. The function in factored form is P(x) 5 22(x 1 3)(x 2 10). c. Determine the vertex of the function. Explain what it means in terms of the problem situation. Then write the function in vertex form. The vertex of the function is (3.5, 84.5). This means that when $3500 is spent on advertising the profit will be at a maximum of $84,500. The function in vertex form is P(x) 5 22(x ) Chapter Assignments 171

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21 Lesson.7 Assignment Name Date More Than Meets the Eye Transformations of Quadratic Functions 1. The owners of a botanical park would like to create a walkway around one of their premier gardens. The garden is 100 feet long and 100 feet wide. The drawing below shows the layout of the garden and walkway. x 100 ft 100 ft a. Determine the function, A(x), that represents the total area of the garden and walkway, Let x represent the width of the walkway. Then, write the quadratic function in vertex form. A(x) 5 (2x 1 100)(2x 1 100) A(x) 5 2(x 1 50)(2)(x 1 50) A(x) 5 4(x 1 50 ) 2 b. Graph the function with the bounds [280, 80] X [210, 1000], with an X-scale of 10 and a Y-scale of 100. Sketch the graph on the coordinate plane provided. Also sketch the graph of the basic function f(x) 5 x 2 on the same coordinate plane. Label the graphs. y f(x) 5 x 2 A(x) 5 4(x 1 50) 2 x Chapter Assignments 173

22 Lesson.7 Assignment page 2 c. Describe how the graph of A(x) compares to the graph of f(x) and define the types of transformations the changes represent. The graph of A(x) is translated 50 units to the left of f(x). This is a horizontal translation. Each y-coordinate of the graph of A(x) is 4 times the y-coordinate of the graph of f(x). This is a vertical dilation with a dilation factor of A physics class has been assigned the task of creating a container that will protect an egg that their teacher will drop from the roof of their school. The graph below shows the basic function f(x) 5 x 2, and also shows the function h(x) which represents the height of the egg with respect to x, the time it is in the air. y h(x) f(x) x 2200 a. Describe the types of transformations performed on f(x) to result in h(x). The graph of f(x) has been reflected about the line y 5 0, has been translated 200 units up, and has been stretched vertically in order to get the graph of h(x). b. If the dilation factor is 16, write the function h(x) that represents the height of the egg. h(x) x Chapter Assignments

23 Lesson.7 Assignment page 3 Name Date 3. A company s revenues are dependent on the amount of product they sell, x. Use the given characteristics to write a function R(x) in vertex form, which represents the company s revenue with respect to x. Then, sketch the graph of R(x) and the basic function f(x) 5 x 2 on the grid. The function is quadratic. The function is continuous. The function has an absolute maximum. The function is translated 70 units up and 100 units to the right from f(x) x 2. The function is vertically stretched with a dilation factor of 1 5. Equation: R(x) (x ) y f(x) 5 x 2 x R(x) (x 2 100) Chapter Assignments 175

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