Answers Investigation ACE Assignment Choices Problem. Core Other Connections, ; Etensions ; unassigned choices from previous problems Problem. Core, 7 Other Applications, ; Connections ; Etensions ; unassigned choices from previous problems Problem. Core,, Other Applications, ; Connections ; Etensions ; unassigned choices from previous problems Problem. Core 7 Other Connections ; Etensions, 7; unassigned choices from previous problems the flare, and the entr for when the height is once again 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. seconds. c. The ball would reach the basket just after. seconds.. a. Height of a Diver After t s Time (t) Height (h) Adapted For suggestions about adapting ACE eercises, see the CMP Special Needs Handbook. Connecting to Prior Units : Covering and Surrounding, Stretching and Shrinking;, : Filling and Wrapping; : Covering and Surrounding;, : Filling and Wrapping;, : Covering and Surrounding Applications. a. At seconds, the flare will have traveled to a maimum height of ft. b. The flare will hit the water when the height is ft, which will occur at s........7.......7..7..7..7... ACE ANSWERS 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.... 7.. Investigation What Is a Quadratic Function?
b. Height (m) Diving From the Platform...... Time (s) 7. 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. (, ) (, ) (, ) O smmetr c. The diver hits the water s surface when the height is, which happens at between and. seconds. In the graph, this is the -intercept. In the table, it is the entr for when height is. 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 between 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. ft, which occurs after about. 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, which occurs at about. seconds. c. The board is 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. 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 Frogs, Fleas, and Painted Cubes... smmetr (, ) (, ) O (, ) (, ) (, ) (, ) smmetr smmetr (, ) (, ) (, )
. 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. 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 -intercepts and minimum at (, ).. We can predict that this is a parabola with -intercepts and maimum at (, ). (, ) smmetr (, ) smmetr. We can predict that this is a parabola with a minimum, and the -intercept at (, ). smmetr Note to the teacher: This graph does not have real roots; that is, it does not cross the -ais. If =, then =-, so is a comple number.. If we factor this we have = ( + ). From this, we can predict this is a parabola with minimum and -intercept at (, -). We can predict the -intercept from = + + ; it is (, ). (, ) (, ) smmetr. We can predict that this is a parabola with a minimum and -intercept at (, ). (., ) (., ) (, ) smmetr ACE ANSWERS Investigation What Is a Quadratic Function? 7
7. We can predict that this is a parabola with -intercepts at and, and a verte at (, ). From the epanded form = we can predict there will be a maimum at (, ).. This is not a quadratic relationship. (Note: If the point (, -) were (, -), this would be a quadratic relationship.). This is a quadratic relationship with a minimum point.. This is a quadratic relationship with a minimum point.. This is not a quadratic relationship. (Note: This has smmetr about the line =, but this has two linear segments; its equation is = «+). This is a quadratic relationship with a minimum point.. a. In each equation, second differences are constant, which means that all the equations are quadratic. The constant second differences for each equation are equal to a, where a is the coefficient of. See tables below. ( ) smmetr (, ) (, ) (, ) Frogs, Fleas, and Painted Cubes. a. 7 7... 7..... b. Since these are quadratic equations, second differences will be constant and will be equal to twice the number multiplied b. For =, second differences will be ; for = a, second differences will be a. c. Yes, the equations are quadratic and the second difference for each is a constant. Table of (, ) Values
b. We know where the maimum point is, so we can find the smmetr and complete the graph b plotting a corresponding point on the right side for each point on the left side.. We know that the minimum point is where =., so we can find the smmetr and complete the graph b plotting a corresponding point on the left side for each point on the right side. (Figure 7). If ou etend the table, ou will get the following values: (-, ), (-, ), (-, ), (-, ), (-, ). Note: The second difference is. 7. a. The corners, or cubes. b. The cubes along the edges that are not corner cubes, or = cubes. c. The large cube has faces, and each face contains = cubes with one face painted, a total of = cubes. d. Removing the eternal cubes leaves =, unpainted cubes. Figure 7 ( ). a. The unpainted cubes form a -b--b- cube on the inside of the large cube, which means the dimensions of the large cube must be -b--b-, with,7 total cubes. 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. c. Each of the edges contains = cubes painted on two faces, which means the large cube must have the dimensions of -b--b-, with,7 total cubes. 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.. 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. 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 7 ACE ANSWERS Investigation What Is a Quadratic Function?
of the number of cubes with faces painted because their third differences are constant.. = ( - ) 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 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.) Connections. a. Table : Each -value is twice the previous -value. The missing entr is (, ). 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. (, ). e. The minimum is not visible in an of the tables, but if the tables are etended, there will be a minimum point.. a. The equations are equivalent. Possible eplanation: When ou graph the equations, the graphs are identical so the equations must be the same, or use the Distributive Propert. 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 for p gives the following values for I: I = ( - p)p = ( - ) = () =,. I = p - p = () - = - =,. I = - p = - = - =-. c. M = ( - p)p -, or M = p - p -. d. A price of $ gives the maimum profit, which is $,. Note: This can be seen in a graph or a table of the equation as shown below.,, Profit From Art Fair Price ($) 7 Profit ($),,7,,,,7, Profit From Art Fair (, ) Price (dollars) e. For prices under about $. and over about $., 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.). a. A = ;P = Frogs, Fleas, and Painted Cubes
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 = () =, so the area would increase b a factor of. Since P = () =, the perimeter would increase b a factor of. d. Since A = m, = m, so P = () = m. e. f. Area Side Length, Perimeter and Area of a Square 7 (, ) 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. = eggs in each laer. b. =,7 eggs in the container.. a. V = b. V = () = ; the volume would increase b a factor of. c. V = () = 7 ; the volume would increase b a factor of 7 and the surface area would increase b a factor of. d. Perimeter Length, Surface Area and Volume of a Solid Edge Length 7 Side Length Surface Area 7 (, ) Volume 7 7,,,7 ACE ANSWERS Investigation What Is a Quadratic Function?
e. Edge Length vs. Surface Area f. The relationship between edge length and surface area appears to be quadratic. The graph looks quadratic, and second differences are constant. (Figure ) The relationship between edge length and volume appears to be some new tpe of relationship because it is not a linear, quadratic, or eponential relation. (Figure ). a. -( - ) = - + b..( - ) = - Edge Length 7. a. ( + )( + ) = + + b. ( + )( + ) = + + Edge Length vs. Volume c. ( - )( - ) = - + The pattern is squaring a binomial, ( + c) when the coefficient of is. The square of a binomial is the square of plus (c)() plus the square of c. Smbolicall this is represented b: ( + c) = ( + c)( + c) = + c + c + c or + c + c. A similar pattern holds when the coefficient of is not. (a + c) = (a + c)(a + c) = Edge Length (a) + ac + ac + c or (a) + ac + c. Surface Area Volume Figure Edge Length (cm) Surface Area (cm ) Figure Edge Length (cm) Volume (cm ) 7 7 7 Third Frogs, Fleas, and Painted Cubes
. a. ( + )( - ) = - b. ( + )( - ) = - c. ( +.)( -.) = -. The pattern is multipling the sum and difference of two numbers. The result is the difference of the squares of the two numbers. Smbolicall, this is represented b: ( + c)( - c) = + c - c - c or - c. A similar pattern holds when the coefficient of is not. (a + c)(a - c) = (a) - c.. a. + + = ( + ) b. - + = ( - ) c. - = ( + )( - ) d. - = ( + )( - ). a. + + = ( + )( + ) b. - = ( + )( - ) c. + + = ( + )( + ). a. The areas are p square units and p square cm. b. The relationship is quadratic. The area increases b increasing amounts. Students might eamine the differences in 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. (Figure ) 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 and -. (Figure ). B. a. (Figure ) 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 square units. Figure Relationship of a Radius to Area of a Circle Radius Area Figure Radius Height Surface Area Surface Areas of Clinders With Different Radius and Height [ ()()] [ ()()] ( )() ( ) ACE ANSWERS Figure Surface Areas of Clinders With Equal Radius and Height Radius Height Surface Area ( )() ( )() ( )() ( )() ( )() Investigation What Is a Quadratic Function?
d. V =. This is not quadratic (it is actuall 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.. No; the surface area of Silvio s bo is, sq. in, since =,. Ten sq. ft. of wrapping paper equals, sq. in since a square foot is square inches and ( in) =, sq. inches. There will not be enough paper.. H 7. C... Building Base Front Right Building Building Etensions.a. If onl the soccer team members go, the cost of the trip is $ per student. The travel agent s profit is the difference between income and cost, or P = n - 7n, where n is the number of students: P = () - 7() =, -, = $,. b. If students go, the cost is $ per student and the agent s profit is P = n - 7n = () - 7() =, -,7 = $,. c. If students go, the cost is $ per student and the agent s profit is P = n - 7n = () - 7() =, -, = $. d. If students go, the cost is $ per student and the agent s profit is P = n - 7n = () - 7() =, -, =-$. For this man students, the travel agent would lose mone.. (Figure ) a. price per student = - (n - ), or - n +, or - n b. income = price n = [ - (n - )]n, or n - n(n - ), or n - n + n, or n - n c. epenses = 7n d. profit = income - epenses = [ - (n - )]n - 7n, or n - n(n - ) - 7n, or -n + n - 7n, or 7n - n. Figure Number of Students Price per Student Pricing and Profit Scenarios for a Travel Agent Travel Agent s Income,,,7, Travel Agent s Epenses 7, 7,7 7, 7,7 Travel Agent s Profit,,,,,7,,7,,,,7, Frogs, Fleas, and Painted Cubes
. a. The agent s profit is greatest for students. b. If fewer than 7 students go on the trip, the agent will make a profit. c. From students to students give the travel agent a profit of at least $,.. a. It takes moves to solve the puzzle with pair of coins. Starting with 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 the moves could be as follows: ACE ANSWERS Investigation What Is a Quadratic Function?
b. (Figure ) c. The numbers of moves calculated from the epression agree with the numbers found above. (Figure ) d. differences are a constant, so the relationship is quadratic. (Figure ). a. The graph of = + is a straight line with slope and -intercept (, ). The graph of = ( + )( + ) is a parabola with a minimum point at (-., -.) and -intercepts at (-, ) and (-, ). The graph of = ( + )( + )( + ) increases as increases, then decreases, then increases again. It has three -intercepts at (-, ), (-, ), (-, ). The graph of = ( + )( + )( + )( + ) is shaped like the letter W. It has two local minimum points, a local maimum point, and four -intercepts at (-, ), (-, ), (-, ), (-, ). Note to teacher: The terms local minimum and local maimum will be introduced in future mathematics courses. The just refer to minimums and maimums over a given part of the graph, which are not necessaril the minimum or maimum for the entire graph. b. The equation = ( + ) has constant first differences. The equation = ( + )( + ) has constant second differences. The equation = ( + )( + )( + ) has constant third differences. The equation = ( + )( + )( + )( + ) has constant fourth differences.. a. blue: ellow: = orange: = b. blue: ellow: = orange: = c. blue: ellow: (n - ) orange: (n - ) (n - ) or (n - ) d. The relationship described b (n - ) is quadratic because it is formed b the product of two linear factors. 7. a. cubes b. cubes c. cubes d. cubes e. + + = cubes, or = cubes Figure # of Tpe of Coin 7 # of Moves Figure Number of Each Tpe of Coin 7 Number of Moves 7 7 Frogs, Fleas, and Painted Cubes
Possible Answers to Mathematical Reflections. Possible situations: (i) the nth triangular number. Question: What is the th triangular number? (ii) the height in a frog jump. Question: What is the highest height in a frog jump? (iii) the number of high fives. Question: How man high fives are there if everone echanges high fives with each other on a team with members? (iv) rectangles with a fied perimeter. Question: What is the greatest area for a rectangle with a fied perimeter of meters?. a. In tables of (, ) values for quadratics the first differences are non-constant but the second differences are constant. b. The graphs of quadratics (if a view window including all four quadrants is used and if a big enough range of values for the - and the -values is used) are parabolas opening upward or downward depending on the sign of the coefficient of the term. The graph has a smmetr through the maimum or minimum point. The smmetr intersects the -ais at the midpoint between the -intercepts. c. The equations that match quadratic relations can be in epanded form or factored form. In the epanded form, = a + b + c, the highest eponent of the independent variable is. If a is positive, then there is a minimum point; if a is negative, then there is a maimum point. The -intercept is c. In a factored form, = ( + b)( + c), there are two factors, each of which has the independent variable raised to the first power. The -intercepts are =-b and =-c.. The patterns of change for linear functions are characterized b constant first differences. The patterns of change for eponential functions are characterized b consecutive differences either increasing without bound or approaching but never achieving it. The patterns of change for quadratic functions are characterized b a constant second difference. Note to the teacher: These are onl general trends and simpl because consecutive differences increase without bound does not impl that a function is eponential. Answers to Looking Back and Looking Ahead. a. All graphs will be parabolas that are concave down with the -intercept and b -intercepts at and and maimum point ( b. At this point in the students, b ) learning about quadratic functions and equations we don t epect such complete abstract reasoning. Most students should know the general shape of the graphs and recognize in general that the maimum point will lie midwa between the two -intercepts. b. Table was produced b the necessar quadratic, while Table shows a constant rate of rise and fall in height, and thus cannot be quadratic. Students might notice that if one looks at the first and second differences of height values in Table, ou get,,, -, -, -, and then -, -, -, -, -. Constant second differences are another indicator of a quadratic relationship.. a. feet. This answer can be found b tracing a table or graph of the height equation. Smbolic reasoning might be used to find that the ball comes back to its starting position when t = and to infer that the maimum point occurs midwa in that time interval. b. seconds. Students might find this result b tracing a table or graph of the equation or b solving with smbolic reasoning. -t + t = -t(t ) = t = and t = c.. seconds and. seconds. Again, students might answer this question b tracing a table or graph of the equation to find -values for which =. We do not reall epect students to solve the equation -t + t = b smbolic reasoning. ACE ANSWERS Investigation What Is a Quadratic Function? 7