Answers Investigation 4

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1 Answers Investigation Applications. a. At seconds, the flare will have traveled to a maimum height of 00 ft. b. The flare will hit the water when the height is 0 ft, which will occur at 0 seconds. c. In a graph, the maimum point represents the maimum height of the flare, and the right-hand -intercept represents the point at which the flare hits the water. In a table, the entr for when the height is its greatest represents the maimum height reached b the flare, and the entr for when the height is once again 0 represents the point at which the flare hits the water.. a. The rocket will travel to a height of feet. It reaches this maimum height after seconds. b. The rocket was launched at a height of feet above ground level. c. It will take seconds for the rocket to return to the height from which it was launched.. a. The ball is released at about. ft (the -intercept). b. The ball reaches its maimum height, about 7. ft, at about 0. seconds. c. The ball would reach the basket just after. seconds.. a. Height of a Diver after t Seconds Time (t) Height (h) b. Diving From the Platform Height (m) Times (s) Frogs, Fleas, and Painted Cubes Investigation Copright Pearson Education, Inc., or its affiliates. All Rights Reserved.

2 Answers Investigation c. The diver hits the water s surface when the height is 0, which happens between and. seconds. In the graph, this is the -intercept. In the table, it is the entr for when height is 0. d. The diver will be m above the water s surface between. and.7 seconds. e. The diver is falling at the greatest rate just before hitting the water s surface. In the table, this is when the difference of successive height values is the greatest. In the graph, this is where the curve has the steepest downward slope.. a. The maimum height is about.0 ft, which occurs after about 0. seconds. Note: Students can find this b making a table or a graph of the equations. b. Her feet hit the water when the height is 0, which occurs at about. seconds. c. The board is 0 ft above the water s surface.. a. The maimum height is ft, which is reached at. seconds. You could find this in a table of time versus height b locating the maimum height. You could find this in a graph b determining the height at the maimum point of the parabola. b. The ball hits the ground just after. seconds. You could find this in a table of time versus height b locating the value for time when height is 0. You could find this in a graph b determining the time at the point at which the parabola crosses the -ais. c. The ball begins rising rapidl and then slows its ascent until it reaches the maimum height of ft. It then starts to fall, slowl at first and gaining speed on the wa down until it hits the ground. d. The ball is ft above ground when thrown = 9 (0, 9) (, 0) (, 0) O 0 0 = smmetr (0, 0) (, 0) O (, ) 0 (0, 0) (, 0) O smmetr = (, 9) smmetr = + + (, 0) (, 0) O (, ) smmetr Frogs, Fleas, and Painted Cubes Investigation Copright Pearson Education, Inc., or its affiliates. All Rights Reserved.

3 Answers Investigation. a. If the sign of the coefficient of the term is negative, the graph will have a maimum point. If it is positive, the graph will have a minimum point. b. The -intercepts are the values that make each factor in the factored form of the equation equal to 0. The -intercept is the constant term in the epanded form of the equation. c. If there are two -intercepts, the distances from each -intercept to the smmetr are the same. If there is onl one intercept, it is on the smmetr. There is not an apparent relationship between the -intercept and the smmetr.. We can predict that this is a parabola with - and -intercepts and minimum at (0, 0). = O smmetr (0, 0). We can predict that this is a parabola with - and -intercepts and maimum at (0, 0). = (0, 0) O 0 smmetr. We can predict that this is a parabola with a minimum, and the -intercept at (0, ). smmetr = + O (0, ) Note: This graph does not have real roots; that is, it does not cross the -ais. If = 0, then = -, so is a comple number.. If we factor = + + 9, we have = ( + ). From this, we can predict this is a parabola with minimum and -intercept at ( -, 0). We can predict the -intercept from = + + 9; it is (0, 9). = smmetr (, 0) O. We can predict that this is a parabola with a minimum and -intercept at (0, -). = smmetr (., 0) (., 0) O (0, ) Frogs, Fleas, and Painted Cubes Investigation Copright Pearson Education, Inc., or its affiliates. All Rights Reserved.

4 Answers Investigation 7. We can predict that this is a parabola with -intercepts at 0 and, and a verte at (, ). From the epanded form = - we can predict there will be a maimum at (, ). smmetr = ( ) (0, 0) O. a. The corners, or cubes. (, ) (, 0) b. The cubes along the edges that are not corner cubes, or * 0 = 0 cubes. c. The large cube has faces, and each face contains 0 * 0 = 00 cubes with one face painted, a total of * 00 = 00 cubes. d. Removing the eternal cubes leaves 0 * 0 * 0 =,000 unpainted cubes. 9. a. The unpainted cubes form a 0-b-0-b-0 cube on the inside of the large cube, which means the dimensions of the large cube must be b b, with,7 total cubes. Of these,,000 have no painted faces, 00 are painted on face, 0 are painted on faces, and are painted on faces. b. Each of the faces on the cube contains = cubes with one face painted. There are cubes arranged in a -b- square, which means the large cube must have the dimensions of b b, with,7 total cubes. Of these,,7 have no painted faces, are painted on face, are painted on faces, are painted on faces. c. Each of the edges contains = cubes painted on two faces, which means the large cube must have the dimensions of b b, with,97 total cubes. Of these,, have paint on no faces, 7 are painted on face, are painted on faces, are painted on faces. d. An cube would have cubes painted on three faces, located at the corners; we cannot tell the size of the large cube based on this information. 0. a. In the values for, first differences are constant. In the values for, second differences are constant. In the values for, the third differences are constant. 9 7 b. In the table of value, the pattern of change is similar to the pattern of the number of cubes with or faces painted because their first differences are constant. In the table of value, the pattern of change is similar to the pattern of the number of cubes with face painted because their second differences are constant. In the table of value, the pattern of change is similar to the pattern of the number of cubes with 0 faces painted because their third differences are constant. Frogs, Fleas, and Painted Cubes Investigation Copright Pearson Education, Inc., or its affiliates. All Rights Reserved.

5 Answers Investigation. = ( - ) is similar to the relationship of the number of cubes painted on two faces because the are both linear. = ( - ) is similar to the relationship of the number of cubes painted on 0 faces or total cubes because the are both cubic. = ( - ) is similar to the relationship for the number of cubes painted on one face because the are both quadratic. Note: Students can observe the similarit from the form of equations or the pattern of changes in tables.. The smmetr can be found b finding the point on the -ais that is halfwa between the -intercepts. If this point is a, then the smmetr is = a. The -intercepts can be read directl from the factored form or estimated from the graph of a quadratic equation.. B; Line of smmetr: = - 9. D; Line of smmetr: = -. C; Line of smmetr: = -. A; Line of smmetr: = 0. a. Possible answer: The both have the same -intercepts and the both have the same ais of smmetr. 7. D b. Possible answer: One opens up and the other opens down. One has a maimum and the other has a minimum. c. (, ); substituting = into = (0 - ) produces =. d. (, -); substituting = into = ( - 0) produces = -. Connections. a. Table : Each -value is twice the previous -value. The missing entr is (, 00). Table : Each -value is greater than the previous -value. The missing entr is (, ). Table : Each increase in the -value is greater than the previous increase. The missing entr is (7, ). Table : Each increase in the -value is less than the previous increase. The missing entr is (, ). b. Table ; = () Table ; = ( + ) Table ; = ( + ) Table ; = - c. Tables and ; in Tables and, the second differences are constant. d. Table ; (0, ). e. None of the tables shows a minimum -value. Because the table has onl a few entries, there is a least -value among those listed. This least -value is not necessaril the least -value of the entire function. Onl the pattern for the function represented in Table has a minimum -value, but it is not listed in the table. Copright Pearson Education, Inc., or its affiliates. All Rights Reserved. 9. a. The equations are equivalent. Possible eplanation: When ou graph the equations, the graphs are identical so the equations must be the same. You can also use the Distributive Propert to show that the equations are equivalent. b. Possible answers: This equation is not equivalent to the other two because its graph is different. Or, substituting the same value for p into all three equations proves that the are not equivalent. For eample, substituting 0 for p gives the following values for I: I = (00 - p)p = (00-0)0 = (0)0 =,00. I = 00p - p = 00(0) - 0 =, =,00. I = 00 - p = 00-0 = = -00. c. M = (00 - p)p - 0, or M = 00p - p - 0. Frogs, Fleas, and Painted Cubes Investigation

6 Answers Investigation d. A price of +0 gives the maimum profit, which is +,0. Note: This can be seen in a graph or a table of the equation as shown below. (See Figure.) Profit From Art Fair Price ($) Profit ($) 0,0,70,00,0,00,70,0 0 e. For prices under about +. and over about +9., the potter will lose mone, so the potter will make a profit on prices between these amounts. Note: These points are the -intercepts; students can approimate them b making a table or a graph. 0. a. * = eggs in each laer. b. * =,7 eggs in the container. a. A = ; P = b. A = () =, so the area would increase b a factor of. P = () =, so the perimeter would increase b a factor of. Note: Students ma solve this b testing several eamples. c. A = () = 9, so the area would increase b a factor of 9. Since P = () =, the perimeter would increase b a factor of. d. Since A = m, = m, so P = () = m. e. Side Length, Perimeter, and Area of a Square Figure,000 Profit From Art Fair (0, 0), Price (dollars) 00 Frogs, Fleas, and Painted Cubes Investigation Copright Pearson Education, Inc., or its affiliates. All Rights Reserved.

7 Answers Investigation f. Area Perimeter (, ) 0 0 Side Length (, ) 0 Side Length g. The relationship is quadratic between the side length () and the area ( ).The relationship is linear between the side length () and the perimeter ().. a. V = b. V = () = ; the volume would increase b a factor of. c. V = () = 7 ; S.A. = () = (9 ) = 9( ); the volume would increase b a factor of 7 and the surface area would increase b a factor of 9. d. e. Volume Length, Surface Area, and Volume of a Cube Edge Length Surface Volume Area ,000 7,,7 Edge Length vs. Surface Area Surface Area Edge Length Edge Length vs. Volume Edge Length 7 Frogs, Fleas, and Painted Cubes Investigation Copright Pearson Education, Inc., or its affiliates. All Rights Reserved.

8 Answers Investigation f. The patterns of change in the table and graph for surface area suggest the relationship between length and area is quadratic. The patterns of change in the table and graph for volume suggest the relationship between length and volume is cubic.. a. The areas are p square cm and p square cm. b. The relationship is quadratic. The area increases b increasing amounts. Students might eamine the differences of areas, or the might graph the radii and area to see if the get a quadratic, or the might use smbols to justif that =. is a quadratic relationship. (See Figure.). B c. The length of the smaller rectangle is the same as the circumference of the smaller circle or p. So the surface area of the smaller clinder is p + p + (p)(), or p square units. The surface area of the larger clinder is p + p + p () or p square units. d. Yes; students might eamine second differences and see that the are a constant p. Or the might identif the equation = p( + ) as the equation of a parabola with -intercepts at 0 and -. (See Figure.). a. (See Figure.) Figure Relationship of a Radius to Area of a Circle Radius Area π π 9π π π Figure Radius Height Surface Areas of Clinders With Different Radius and Height Surface Area π π [ ()()]π = 0π [ + + ()()]π = π π + π + (π)() = π + π = π( + ) Figure Radius Height Surface Area Surface Areas of Clinders With Equal Radius and Height π + π + (π)() = π π + π + (π)() = π 9π + 9π + (π)() = π π + π + (π)() = π π + π + (π)() = π Frogs, Fleas, and Painted Cubes Investigation Copright Pearson Education, Inc., or its affiliates. All Rights Reserved.

9 Answers Investigation b. Yes; students might eamine the second differences and see that the are a constant p, or the might identif the relationship s equation of = p as the equation of a parabola.. a. Each edge is units. b. The surface area is square units. The volume is 7 cubic units. c. Student drawings should show the flat pattern of a cube with edge units, surface area () or 9 square units. There are man such nets. One possibilit is shown... Building Building. If =, then ( - ) = 0. If = -, then ( - ) = 0.. If =, then - =. If =, then - = 0.. If =, then + + =. If = -, then + + = 0.. If = -, then ( - 7)( + ) = 0. If =, then ( - 7)( + ) = -0. d. V =.This is not quadratic (it is a cubic relationship). Students might make a table and eamine how the volume grows, or the might graph = and eamine the shape, or the might refer to the smbols. 7. No; the surface area of Silvio s bo is, square inches, since * =,. Ten square feet of wrapping paper equals,0 square inches since a square foot is square inches and 0() =,0. There will not be enough paper.. H 9. C 0. Building Base 7. matches Graph 9 and Propert Card matches Graph and Propert Card or matches Graph 0 and Propert Card matches Graph and Propert Card 7 matches Graph and Propert Card or matches Graph 7 and Propert Card Note: It ma be worth discussing which properties are sufficient to make a parabola unique. For eample, Graphs and have the same -intercept, same -intercepts, and the same smmetr, but the are different graphs. Had the smmetr been replaced with the verte (ma. or min. point), it would be enough to make the parabola unique.. a. The equation was = ( + ). The new graph would have -intercepts at = - and = -, so the equation would be = ( + )( + ). b. The equation was = ( + ). The new graph would have -intercepts at = 0 and =, so the equation would be = ( - ). Front Right 9 Frogs, Fleas, and Painted Cubes Investigation Copright Pearson Education, Inc., or its affiliates. All Rights Reserved.

10 Answers Investigation c. If ou translate parabola G7 to the right b units, its -intercepts would coincide with those of parabola G. However, their orientations are different: parabola G7 opens upward, while parabola G opens downward. So, ou would also need to reflect parabola G7 over the -ais in order to transform it into parabola G. Etensions 9. a. It takes moves to solve the puzzle with pair of coins. Starting with the moves could be as follows: It takes moves to solve the puzzle with pairs of coins. Starting with the moves could be as follows: the moves could be as follows: It takes moves to solve the puzzle with pairs of coins. Starting with 0 Frogs, Fleas, and Painted Cubes Investigation Copright Pearson Education, Inc., or its affiliates. All Rights Reserved.

11 Answers Investigation b. (See Figure.) c. The number of moves calculated from the epression agree with the numbers found above. (See Figure 7.) d. Second differences are a constant (See Figure.), so the relationship is quadratic. 0. a. If onl the 0 soccer team members go, the cost of the trip is + per student. The travel agent s profit is the difference of income and cost, or P = n - 7n, where n is the number of students: P = (0) - 7(0) =,00 -,00 = +,000. b. If students go, the cost is +0 per student and the agent s profit is P = 0n - 7n = 0() - 7( =,000 -,7 = +,. c. If 0 students go, the cost is + per student and the agent s profit is P = n - 7n = (0) - 7(0) =,00 -,00 = +00. d. If 0 students go, the cost is + per student and the agent s profit is P = n - 7n = (0) - 7(0) =,00 -,000 = For this man students, the travel agent would lose mone. Figure Number of Each Tpe of Coin Number of Moves First Differences Second Differences Figure 7 # of Tpe of Coin # of Moves Frogs, Fleas, and Painted Cubes Investigation Copright Pearson Education, Inc., or its affiliates. All Rights Reserved.

12 Answers Investigation. (See Figure.) a. price per student = - (n - 0), or - n + 0, or - n b. income = price * n = [ - (n - 0)]n, or n - n(n - 0), or n - n + 0n, or n - n c. epenses = 7n d. profit = income - epenses = [ - (n - 0)]n - 7n, or n - n(n - 0) - 7n, or -n + n + 0n - 7n, or 70n - n.. a. The agent s profit is greatest for students. b. If fewer than 70 students go on the trip, the agent will make a profit. c. From 0 students to 0 students give the travel agent a profit of at least +,00.. a. blue: ellow: * = 9 red: * = b. blue: ellow: * = red: * = c. blue: ellow: (n - ) red: (n - ) d. The relationship described b S = (n - ) is quadratic because it is formed b the product of two linear factors. The relationship S = (n - ) is linear.. a. 0 cubes b. cubes c. cubes d. cubes e. + + = 0 cubes, or * * = 0 cubes Figure Pricing and Profit Scenarios for a Travel Agent Number of Students Price per Student Travel Agent s Income 0 0 =,00 =,0 =,70 =,0 Travel Agent s Epenses 0 7 =,00 7 =,7 7 =,0 7 =,7 Travel Agent s Profits,00,00 =,000,0,7 =,09,70,0 =,0,0,7 =,0 Frogs, Fleas, and Painted Cubes Investigation Copright Pearson Education, Inc., or its affiliates. All Rights Reserved.