UNIT TWO QUADRATIC FUNCTIONS 35 HOURS MATH 521A

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1 UNIT TWO QUADRATIC FUNCTIONS 35 HOURS MATH 51A Revised March 0, 01 31

2 Quadratic Function Introductory Lesson An excellent way to begin the study of quadratics is by looking at a real-world problem where quadratics comes into play. A publishing company finds that it can produce a particular book for $15 each plus initial setup costs of $10,000. Complete the following table to determine the profit made by the company. On the TI-83 graph profit vs book price and answer the following questions: a) What basic law of economics does the graph illustrate? b) What is the optimum price for the company to sell the books for so that profits are maximized? Estimate the maximum from the graph then use the trace feature on the TI-83 to confirm your estimate. # books sold Total Printing costs #books $ Selling Price Total Sales Revenue Profit 1100 $ $ $ $ $ $ $ $ $ $ $10 This graph is a type of quadratic function called a parabola. and describes many phenomena in the real world. It finds application in a myriad of fields such as business, architecture, the social sciences and science. Throw a baseball from left field to home plate and it travels a quadratic path. The roofs of many buildings (Skydome for example) are quadratic in shape. Arches in bridges are quadratic due to their structural integrity. Car headlight reflectors are quadratic. Satellite dishes are quadratic. The list is endless. 3

3 Solution: a) In economics as price increases demand for that product decreases. b) From the graph an estimate of $70 seems to optimize profits. Note to Teachers: If a quadratic regression is performed and the maximum is found we see that the exact maximum is when the price is $

4 SCO: By the end of grade 11, students will be expected to: E3 use transformations to draw graphs C5 construct and analyze graphs and tables relating two variables Elaborations - Instructions Strategies/Suggestions Graphing y = ax + q (3.1) Challenge student groups to do the Warm Up and Mental Math exercises listed in the Suggested Resources. Do the Investigation on p Introduce students to the concept of a Mathematical Graphing Ladder : < Rung 1) 1 st degree (linear) functions yielding a straight line graph. < Rung ) nd degree (quadratic) functions yielding a parabolic curve. Invite students to generate a table of values and graph y = x on the TI- 83 and manually. C60 describe quadratic, exponential and logarithmic relationships and translate among the various representations (graphs, tables of values and written descriptions) Allow student groups time to examine and discuss examples on p Students should become familiar with the formula: y = a x + q a = produces a vertical stretch (y values are changed by a factor of a ). q = produces a vertical translation of the basic function Demonstrates the effect of a on the basic function Demonstrates the effect of q on the basic function 34

5 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Graphing y = ax + q (3.1) Group Activity Generate a table of values for y = x and sketch the graph. Check your work on the TI-83. Group Activity Generate a table of values for y =!x and sketch the graph. Check your work on the TI-83. Pencil/Paper/Technology Determine tables of values and draw graphs for y = x, y = x + and y = x! 3. Check your work on the TI-83. What transformation has q performed on the basic function? Graphing y = ax + q (3.1) Math Power 11 Warm Up p.101 #1-8 Mental Math p.101 #1-8 Investigation p.10 Green # 1-6 Math Power 11 p.109 # 13-19, 8-31 See y = a x + q worksheet at end of unit Journal Write to explain how y = x! is a transformation of y = x. Pencil/Paper/Technology Determine tables of values and draw graphs for y = x, y = x, y = 3x and y = ¼ x. Check your work on the TI- 83. Write to explain the effect of a on the basic graph. Communication Write to explain the transformation of y = x! 3 from the basic y = x function. Use a TI-83 to confirm your prediction. Share your results with the other groups in the class. Pencil/Paper/Technology Determine tables of values and draw graphs for y = x, y = x + 1, y = ¼ x!. Check your work on the TI-83. What transformations have a and q performed on the basic function? 35

6 SCO: By the end of grade 11, students will be expected to: C4 evaluate and interpret non- linear equations using graphing technology C83 determine and interpret the maximum and minimum points of a graph Elaborations - Instructions Strategies/Suggestions Graphing y = ax + q (3.1) Challenge student groups to re-examine exercises 1-3 on p and discuss the concepts of: < axis of symmetry < direction of the opening < coordinates of the vertex < the maximum or minimum < the x-intercepts ( zeros) < the y-intercepts < the domain and range Allow students to read and discuss ex. 4 on p.107. The equation for a parabola can be determined here if you are given: < the vertex and the a value < the vertex and another point on the parabola Students should appreciate that the x-intercepts are called the zeros of the function and occur when the y value or function value is zero. The vertex is equidistant (horizontally) from the x-intercepts. Note to Teachers: If you have students asking: When will we ever use this? look at the Quadratic Functions Applications Worksheet at the end of the unit. 36

7 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Graphing y = ax + q (3.1) Activity Sketch the graph of y = x! 1 and determine: a) the direction of the opening b) the coordinates of the vertex c) the equation of the axis of symmetry d) the domain and range e) the maximum or minimum f) the intercepts Graphing y = ax + q (3.1) Math Power 11 p.109 #1-1 odd, 3-34, 39-4, ,53-55,57,58, 60,6,64-66,69,70,74 Pencil/Paper Write an equation for a parabola with: a) vertex (0,!3) and passing through (!,4) b) vertex (0,), a =!3 See Quadratic Applications Worksheet at end of unit Problem Solving Strategies Math Power 11 p.113 #1,,6,9 37

8 SCO: By the end of grade 11, students will be expected to: E3 use transformations to draw graphs C4 evaluate and interpret non- linear equations using graphing technology C83 determine and interpret the maximum and minimum points of a graph Elaborations - Instructions Strategies/Suggestions Graphing y = a(x! p ) + q (3.3) Invite student groups to read and discuss p and do the Explore activity on p.114,115. Students should appreciate the standard form (vertex form) of a quadratic function: y = a(x! p) + q a = vertical stretch factor p = the horizontal translation q = the vertical translation where (p,q) is the vertex Students should examine ex. 4 on p.117 so that they may understand how to determine the equation of a parabola. A quadratic function can also be written in factored form: y = a (x! z 1 )(x! z ) C47 investigate and articulate how various changes in the parameters of an equation affect the graph and use the results to find the equation of a given graph where z 1 and z are the zeros of the function. 38

9 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Graphing y = a(x! p ) + q (3.3) Pencil/Paper Sketch the graph of each parabola and state: a) the direction of the opening b) the coordinates of the vertex c) the maximum or minimum d) the equation of the axis of symmetry e) the domain and range i) y = ( x + 1) + 3 ii) y = 3( x + 1) iii) y = ( x + 1) Technology Determine any x and y-intercepts using the TI-83 for: a) y = ( x + 1) 3 b) y = ( x 1) + 1 c) y = 3( x + ) 4 Graphing y = a(x! p ) + q (3.3) Math Power 11 p.118 #1-14,5-7, 9,3-37,47-69 odd Applications p.119 #70-75,78,79,81 See Quadratic Application Worksheet at end of unit See y = a(x! p) + q worksheet at end of unit Problem Solving Strategies p.13 #,3,5 Pencil/Paper Write an equation of the parabola with: a) vertex (,!3), y-intercept! b) zeros (x-intercepts) of! and 4, passing through (1,) 39

10 SCO: By the end of grade 11, students will be expected to: Elaborations - Instructions Strategies/Suggestions Completing the Square (3.5) Allow student groups to read the introduction on p16 and do the Explore and Inquire on p16-7. C6 develop and apply strategies for solving problems Students should be able to change a quadratic function from: General Form y = ax + bx + c to Standard Form y = a(x! p) + q and back again C54 model real-world phenomena with linear or quadratic equations Challenge student groups to read and discuss examples 1-4 on p Students should be able to appreciate what the important features of the standard form of a quadratic function mean. Ensure a complete discussion about example 4 and the interpretation of the results. Note to Teachers: There are a number of good examples on the worksheet at the end of the unit. If some of them are assigned to the students, then perhaps the students could present their solution and interpretation to the class. It is important that students see the graphs as well and have the numbers in the general and standard forms related to specific points on the graph. 40

11 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Completing the Square (3.5) Presentation Complete the square for the following problem and explain the advantages of a function written in general form and in standard form. A baseball is thrown from the top of a building and falls to the ground below. Its path is approximated by the equation y =!5t + 5t a) how tall is the building? b) when does the ball reach its maximum height? c) how high above the building is the ball at its maximum height? d) when will the ball hit the ground? Solution Completing the Square (3.5) Investigation p.15 Math Power 11 p.131 #1,5,11,13,8, 10,17-1,3-5, 9-55 odd Applications p.13 # 71-73,76,77, 81,83-85,87 See Applications Worksheet at end of unit Students need to see a picture of the path of the ball and realize that the: < general form tells them the height of the ball when it is released is 30 m ( the y-intercept) < standard form h = 5( t 05. ) tells them the vertex is (0.5,31.5) which means that the ball reaches a maximum height of 31.5 m ( 1.5 m above the building) 0.5s after it is released. Pencil/Paper Find the value of c that will make x + 0x + c a perfect square trinomial. Journal Write to explain how to convert a trinomial into a perfect square trinomial. 41

12 SCO: By the end of grade 11, students will be expected to: C3 solve problems using graphing technology C19 graph equations and/or inequalities and analyze graphs both with and without graphing technology Elaborations - Instructions Strategies/Suggestions Solving Quadratic Equations by Graphing(4.1) Invite student groups to read and discuss p.15 and do the Explore and Inquire exercises on that page. Students should come to appreciate that the x-intercepts of a quadratic function correspond to the zeros or real roots of the related quadratic equation. Student groups should read and discuss p Conclusions evolving from the discussions should be that graphing with the TI-83 is a simple and efficient means of determining the nature of the roots. In other words, within moments, a student can enter the quadratic function into the TI-83 and view the graph. Real Roots( x-intercepts) Real and Equal Roots y = x! x! 3 y = x! 4x + 4 no Real Roots y = x + x + 3 In real world applications, using technology (TI-83) is the easiest and most logical method of solving quadratic equations. In most real situations, the roots (if there are any) are not simple integers but are in fact rational numbers: Ex. y = x!3x! 5 where the roots are! and

13 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Solving Quadratic Equations by Graphing(4.1) Technology Solve by graphing a related function: a) x! 3x! 7 b) x! 5x + 1 Technology The function h(d) =!0.0 d + d models the height, h(d) in metres, of a field goal attempt in a CFL football game. The uprights are 46 m away from where the ball is tied. a) Will the ball travel far enough to reach the uprights in the end-zone? b) If the horizontal bar on the uprights is 3 m high, will the ball be high enough to go over the uprights? c) At what height is the ball when and if it goes over the uprights? Solving Quadratic Equations by Graphing(4.1) Math Power 11 p.155 #1,3,4,5,16,17, 19,,9,31,35 Applications p.156 #41,4,44-46,49,5 43

14 SCO: By the end of grade 11, students will be expected to: Elaborations - Instructions Strategies/Suggestions Solving by Factoring (4.) Invite student groups to read and discuss p.157 and do the Explore and Inquire exercises. Students should appreciate that this method works for only very simple quadratic equations that are can be factored. C0 solve quadratic equations by factoring Challenge student groups to examine Ex.1,,3,5, & 6 on p Note to Teachers: Students will not look at Ex. 4 on p.159. This work is in Math 51B. 44

15 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Solving by Factoring (4.) Pencil/Paper Solve and check: a) c! 4c! 1 = 0 b) n! 7n = 15 Solving by Factoring (4.) Math Power 11 p.160 #1,3,5,7, 1-39 odd, 44,63,66,70 Applications p.161 # 75,79,80 Activity Write a quadratic equation with the given roots: a)!5, b) 1/3,! 3/4 c) 4,! /5 45

16 SCO: By the end of grade 11, students will be expected to: A3 demonstrate an understanding of the role of irrational numbers in applications A4 approximate square roots B9 use the calculator correctly and efficiently for various computations Elaborations - Instructions Strategies/Suggestions What is a radical? Invite a short discussion by student groups on the term radical. Ideas they come up with could be listed on the board. These might include: < a radical is a compact way of writing an irrational number similar to scientific notation being a compact and convenient way of writing very small or very large numbers. < parts of a radical are! radical symbol! radicand! index (degree of the root) Index º radical symbol 9 34» radicand 3 Estimation of Radicals Encourage student groups to use guess and check to estimate the values of various degree roots and check their accuracy using a scientific calculator. Simplifying Radicals entire to mixed Challenge students to reduce the size of the radicand as much as possible and still have an equivalent radical expression. The idea is to find two factors of the radicand, one of which is the highest possible perfect square (cube, etc.). Ex. 48 = 16 3 = 4 3 For even degree roots of variable radicands the concept of principal root must be considered. Example: the square root of 16 can be either ± 4 The symbol means principal square root (or positive root) ˆ 16 = 4 and not! 4 Rationalizing One Term Denominators A brief introduction to this topic is needed here. 46

17 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources What is a radical? Journal Write a short explanation of the advantages of writing numbers in radical form. Estimation of Radicals Estimation/Technology Use guess and check to approximate, to the nearest hundredth, the value of the following. Then use a calculator to check the accuracy of your guess. a) 40 3 b) 9 4 c) 0 5 d) 00 Technology Use decimal approximations to arrange the following from smallest to largest. 15,, 5, 4,.3 Simplifying Radicals entire to mixed Pencil/Paper Write each of the following in simplest form. 1) 96 ) 3 68x What is a radical? Estimation of Radicals Math Power 10 p.14 # 3-17 odd Math Power 10 p.15 # Simplifying Radicals entire to mixed Math Power 10 p.19 # 1-15 odd Algebra, Structure and Method Book p.68 # 39,40 Applications Math Power 10 p.0 # 56,57 Activity Rationalize the following denominators: 3 a) b) 5 Journal What advantage is there to rationalizing the denominator? 47

18 SCO: By the end of grade 11, students will be expected to: Elaborations - Instructions Strategies/Suggestions The Square Root Principle (4.3) Invite student groups to read and discuss p and do the Explore and Inquire on p.163. C6 develop and apply strategies for solving problems As a result of discussing the examples on p.164-5, student groups should be able to understand the need to find either: < the exact roots < solutions rounded to a certain number of decimals Ex. 3 on p.165 is a precursor to the next section where the second last line in solving by completing the square looks like the beginning of Ex. 3 on p

19 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources The Square Root Principle (4.3) Pencil/Paper Solve: 4b = 49 Pencil/Paper Solve and check: 3x 48 = 0 The Square Root Principle (4.3) Math Power 11 p.165 #1,3,5,9,13,15, 7,9,46,51, 53,61,63 Applications p.166 #71,83,84 Technology Find the roots to the nearest hundredth: 5x 3 = 19 Group Activity Find the exact solutions for: 7 = 3 + y. Pencil/Paper Solve and check: ( x 1) = 49. Pencil/Paper/Technology Find the exact solutions, then calculate to the nearest hundredth: x 9 = 0. 49

20 SCO: By the end of grade 11, students will be expected to: Elaborations - Instructions Strategies/Suggestions Completing the Square (4.4) Student groups should do the Explore and Inquire on p C6 develop and apply strategies for solving problems They should then look at example 1 on p.169 and in groups come to a consensus on a procedure for solving quadratic equations by completing the square ( making one side a perfect square trinomial). Their study should then extend to examples,3 and 4 on p where they should summarize the procedure for completing the square with quadratics that have a 1. 50

21 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Completing the Square (4.4) Pencil/Paper Solve by completing the square. Express the solution in simplest radical form: a) x + 8x + 4 = 0 b) 6a + 3a = 0 Activity The Jumbotron at the Sky Dome is a rectangle with a perimeter of 87. m and an area of 336 m. What are the dimensions of the screen? Completing the Square (4.4) Math Power 11 p.17 #1,5,13,15,19, 7,30,35,39, 47,50 Applications p.17 #55,65,68 51

22 SCO: By the end of grade 11, students will be expected to: Elaborations - Instructions Strategies/Suggestions The Quadratic Formula (4.5) Invite student groups to read and discuss the derivation of the quadratic formula on p.174. C6 develop and apply strategies for solving problems The roots for the sky-jumper example are on the top of p.175. Challenge student groups to work out the answers to the nearest tenth of a metre and discuss the plausibility of the answers. B31 derive and apply the quadratic formula C101 solve quadratic equations using the quadratic formula Student groups should do the Inquire questions on p.175. b b ac The quadratic formula x = ± 4 seems a little a daunting but actually once students have done a number of problems the formula becomes ingrained in their memory. Remind students that they can always derive the formula, if necessary. Allow time for student groups to read and discuss Ex. 1, & 5. Note to Teachers: Students will not be expected to solve rational equations like Ex. 3 & 4 until Math 51B. Note to Teachers: Invite students to study the program Quadratic. You may want to put into your calculator and link it to one student calculator and then let the students link the program( It will spread geometrically). You might also want to show the overhead of the program and allow time for student groups to try their hand at programming their own calculators. 5

23 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources The Quadratic Formula (4.5) Pencil/Paper Solve using the quadratic formula and verify: a) x 3x + 1 = 0 b) x 4x + 1 = 0 Pencil/Paper Solve using the quadratic formula: 5y 6y 3 = 0. Express answers in simplest radical form. The Quadratic Formula (4.5) Math Power 11 p.178 # 1,5,9,13,15, 1,5,3,41 Applications p.179 #59,61 Technology Solve using the TI-83 program Quadratic : x + 3x = 5x 1 Journal Write to explain what the letters a, b and c in the quadratic formula represent. Group Presentation Choose one of the methods used to solve quadratic equations and describe the advantages and disadvantages of that method. 53

24 SCO: By the end of grade 11, students will be expected to: B43 simplify and perform operations on complex numbers B3 demonstrate an understanding of the role of imaginary numbers as roots of quadratic equations Elaborations - Instructions Strategies/Suggestions Complex Numbers (4.6) Complex numbers are used in many fields: electromagnetic theory, where the electric and magnetic components must be plotted using complex numbers. This spawns innumerable practical applications such as HDTV or any other application where e-m radiation is employed. The theory behind the design and construction of these devices must use complex numbers. Invite students groups to read and discuss the top half of p.181 and do the Explore and Inquire exercises. Student groups should read and discuss the top half of p.18. Teachers may have to guide students through the ideas here. The set of complex numbers contains: < all previously known sets of numbers < pure imaginary numbers of the form bi (5i,!i, i 3 ) imaginary numbers of the form a + bi ( + 3i,! + i) Student groups should read and discuss Ex. 1, & 3 on p They should become familiar with writing complex numbers and doing operations with them. Student groups should read and discuss p.183 & Ex.4,5 & 6 on the following pages. It should be understood that when an equation yields complex roots, the graphical interpretation is that there is (are) no x- intercept(s). Extension: Students could examine Ex. 7 on p.185 and do the exercise at the end of the unit on fractals. A teacher could talk about Siepinski s Triangle and show the TI-83 program on transparencies and the accompanying graph this program produces. 54

25 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Complex Numbers (4.6) Pencil/Paper Simplify: a) 36 e) ( + 3i) + ( 3 5i) b) 7x 3 f) ( 4 i) ( i) c) i 7 g) 35 ( i) d) i 100 h) ( 3i)( 4 + 7i) Complex Numbers (4.6) Math Power 11 p.185 # 1-54 odd, odd,88 Pencil/Paper Solve: a) x x + 5 = 0 b) 3x + x + 4 = 0 Group Activity Do the exercise on constructing Siepinski s Triangle manually. To construct Siepinski s Triangle manually go to: npr.html Group Activity/Technology Enter the Siepinski s Triangle program into your TI-83 and the calculator construct the fractal. 55

26 SCO: By the end of grade 11, students will be expected to: Elaborations - Instructions Strategies/Suggestions Discriminant (4.7) Invite student groups to read and discuss p The discriminant, b! 4ac, is that part of the quadratic formula that determines the type (or nature) of the roots. C61 describe and apply the characteristics of quadratic relationships < b! 4ac real and distinct roots < b! 4ac real and equal roots < b! 4ac imaginary roots 56

27 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Discriminant (4.7) Pencil/Paper Determine the nature of the roots: a) 3x x + 1 = 0 b) ( x 3) = x Discriminant (4.7) Math Power 11 p.189 #1,5,9,13,17, 1,5,7,31, 33,35,38,40 Journal Write to explain how to determine, without graphing, the number of points in which each function intersects the x-axis: a) f ( x) = x + x + 3 b) f ( x) = x 5x 1 c) f ( x) = 4x + 4x + 1 Activity Determine the value(s) of k that yield(s) the type of solution(s) indicated: a) x + 6x k ; distinct real roots b) 3x kx + 4 ; imaginary roots 57

28 Siepinski s Triangle Fractal Activity Use triangular grid paper: Step one: Draw an equilateral triangle with sides of units each. Connect the midpoints of each side. Shade out the triangle in the centre. Step two: Draw an equilateral triangle with sides of 4 units each. Connect the midpoints of the sides and shade the triangle in the centre as before. Step three: Draw an equilateral triangle with sides of 8 units each. Follow the same procedure as before, making sure to follow the shading pattern. Step four: Draw an equilateral triangle with sides of 16 units each. Follow the same shading pattern as before. 58

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30 Sierpinski s Triangle program The program may take minutes to run on the TI-83. Once it has developed the picture, any time you want to see the picture: nd draw cursor over to sto :Recall Pic press enter then type in 6 and press enter. 60

31 y = a x + q Investigation 1. Explore the role of a using the TI-83. Graph the following on the same screen. Then sketch approximate graphs on the graph paper below. a) y = x b) y = x c) y = 4x d) y = 1 x e) y = 1 x 4 f) What effect does a have on the basic function?. Explore the role of q using the TI-83. Graph the following on the same screen. Then sketch approximate graphs on the graph paper below. a) y = x b) y = x + c) y = x + d) y = x e) y = x 4 4 f) What effect does a have on the basic function? 61

32 y = a(x! p) + q Investigation 1. Explore the role of p using the TI-83. Graph the following on the screen. Then sketch approximate graphs on the graph paper below. a) y = x b) y = ( x ) c) y = ( x 4) d) y = ( x + ) e) y = ( x + 4) f) What effect does p have on the basic function?. Determine the equations of the following graphs (note: with the grid on we see the window dimensions): a) c) b) d) Quadratic Functions Applications Worksheet 6

33 Factored Form of a Quadratic Function (3.3) 1. A car driver applies the brakes and skids through an intersection. The investigating officer knows that the distance of the skid depends on the speed of the car. The officer uses the following chart(for the type of vehicle involved) to determine the car s speed before the brakes were applied. Speed(km/h) Length of skid (m) a) Draw the curve of best fit for this data.(use the TI-83). b) Use the curve to estimate the initial speed of the car if the skid mark is 104 m long. (Hint: Use the TI-83 to get the equation that best describes the data and extrapolate to get the solution.). A family wants to make the largest possible rectangular garden possible using 18 m of fencing. The garden is going to back on the barn and therefore only needs fencing on three sides. Determine the dimensions that maximize the area of the garden. Solution: The x- coor dinate of the vertex is halfway between the zeros at x = 4.5. The maximum area enclosed by the fence is 40.5 m when the width is 4.5 m and the length is 9 m. 63

34 3. A style of T-shirt sells for $10. At that price the store can sell 30 a week. The store finds that for every $.1 change in price there is a change in sales of 1 T-shirt ( as price goes up sales go down). a) Set up a table of values that shows how unit price affects sales and total revenue. b) Write a quadratic formula that models this situation. Graph this and explain why there are two points on the graph where the revenue is $0. Why does only one of these points make sense? c) What is the number of T-shirts that should be sold in order to maximize revenue? Solution: The y-axis here represents total revenue. The first graph shows the initial situation where 30 T-shirts are sold for $10 each. Looking at the bottom 3 graphs from left to right: 1) the left graph shows that after 100 price decreases the sales price of the shirt is x = $0 and hence revenue drops to zero. ) the next graph is where there has been 35 price decreases (price is x = $6.50) and the number of shirts sold is 30 + x = 65 yielding a total revenue of $4.50 which is a maximum. 3) the situation here is that when there has been 30 price increases the price is x = $13 and the number of T-shirts sold 30! x goes to zero and hence the y value (revenue) is zero. Apparently at $13 no one will buy the shirts. 4. A music store can sell 80 CD s at $0 each. For every $.50 increase in unit price the sales will drop by 5 CD s. What unit price will maximize the revenue? The same interpretation can be applied here as in the previous problem. 64

35 In the last two problems we can see that the y-intercept represents the original situation before any changes have been made to the price. The x-intercepts represent the situations where the y-value( the revenue) has gone to zero; hence the term zeros of the function. 5. A golf ball is hit and its height is modelled by h = 9.4 t! 4.9 t where h is the height in metres and t is the time in seconds. a) when is the golf on the ground? b) at what time does the golf ball reach its maximum height? c) what is the ball s maximum height? Solution: The equation could be factored to yield: h =!4.9 t(t! 6) which means that the ball is on the ground (h = 0) when t = 0 or t = 6 s. Halfway between these zeros the ball should be at its maximum height (ie. t = 3 s ). Find the ball s maximum height by substituting t = 3 s into the original equation. 6. The data represents the flight of a glider launched from a tower on a hilltop. The height values are negative whenever the glider was below the height of the hilltop. a) how tall is the tower? b) find an equation to model the flight path of the glider. c) find the lowest point in the glider s path. Time (s) Height (m) Time (s) Height (m) ! ! ! ! ! ! ! !3.5 65

36 6. (cont d.) Here the y-axis represents the height and the x-axis represents the time in air. a) When t (x) = 0 the glider hasn t been released yet so thus from the second graph we see the glider is still being held at the top of the tower which is at a height of 9 m. We could use regression analysis on the calculator to get the equation but instead use the factored form of a quadratic function. Find the zeros of the graph; they will turn out to be at t = 3 and at t = 1. Substitute these into the factored form: y = a( x z1)( x z) h = a( t 3)( t 1) substituting ( 09, ) for ( t, h) 9 = a( 0 3)( 0 1) 1 4 = a h = 1 ( t 3)( t 1) 4 factored form h = 1 15 t t

37 7. A golfer makes a perfect shot into the hole 100 m away. At the top of the ball s path it is 10 m above the ground. Assuming the ground from the golfer to the hole is perfectly level, find an equation to model the path of the ball. What is the height of the ball when it is 15 m from the hole? 8. The underside of a bridge has the shape of a parabolic arch. It has a maximum height of 30 m and a width of 50 m. Can a sailboat with a mast 7 m above the water pass under the bridge at a distance of 8 m from the axis of symmetry of the arch? Justify your answer. Standard Form (Vertex Form) of a Parabola (3.3 cont d.) 9. A ball is hit into the air. Its height H ( in metres) after t seconds is H =!5(t! 4) + 10 a) In which direction does the parabola open? How do you know? b) What are the coordinates of the vertex? What does it represent? c) From what height was the ball hit? d) Find one other point on the curve and interpret its meaning. 10. When a stationary object is released to fall freely, its height h, in metres, after t seconds is h =!4.9 t + k. For a charity event, the principal pays to drop a watermelon from a height of 100 m. a) The clock that times the fall of the watermelon runs for 3 s before the principal releases the watermelon. How does this affect the equation and the graph? What is the new equation? b) Now, suppose the principal drops another watermelon from a height of 50 m. How does the graph change? What is the new equation? 11. When a baseball is hit, its height h in metres, after t seconds is h =!5.(t! 3) a) what is the height of the ball at the instant it is hit? b) find the maximum height of the ball and the time when this height is reached? c) the ball hits the outfield wall at t = 5.9 s. How high up the wall does the ball hit? 1. A rock is tossed into the air from a bridge over a river. If its height h above the water, in metres, after t seconds is h =!4.9 (t! ) + 9; a) from what height above the water is the rock tossed? b) find the maximum height of the rock and the time when this height is reached. c) is the ball still in the air after 4.5 s? Solve by Completing the Square (3.5) 13.A baseball is thrown from the top of a building and falls to the ground below. Its path is approximated by the relation h =!5 t + 5 t a) how tall is the building? b) when will the ball hit the ground? c) when does the ball reach its maximum height? d) how high above the building is the ball at its maximum height? 67

38 14. A model rocket is shot into the air and its path is approximated by h =! 5 t + 30 t, where h is the height of the rocket above the ground in metres and t is the elapsed time in seconds. a) when will the rocket hit the ground? b) what is the maximum height of the rocket? We can see that the zeros are before the rocket is shot and at a time of t = 6 seconds. Half-way between the zeros (t = 3 seconds) should be where the rocket reaches its maximum height. If we substitute this t value into the modelling equation we get h = 45 m. 15. A company that manufactures snowboards uses the function P = 16 x! 81 x to model its profit. In the model, x represents the number of snowboards in thousands and P represents the profit in thousands of dollars. a) what is the maximum profit the company can earn? b) how many snowboards must it produce to earn this profit? 16. A computer software company models the profit on its latest game using the function P =! x + 8x! 90, where x is the number of games it produces in hundred thousands and P is the profit in millions of dollars. a) what is the maximum profit the company can earn? b) how many games must it produce to earn this profit? c) what are the break-even points for the company? 17. A company that makes glassware has a daily production cost C in dollars described by C = 0. b! 10 b + 650, where b is the number of pieces of glassware made. How many items can be made in order to minimize production cost? What is the cost when this many items are made? 18. A model rocket is launched straight upward with an initial velocity of 00 m/s. The height of the rocket h, in metres, can be modelled by h =! 5 t + 00 t where t is the elapsed time in seconds. What is the maximum height the rocket reaches? 19. The cost C, in dollars, of operating a concrete-cutting machine is modelled by C =. n! 66 n + 655, where n is the number of minutes the machine is run. How long must the machine for the operating cost to be at a minimum? What is the minimum cost? 0. Find the dimensions of a rectangle that has a perimeter of 40 cm and the largest possible area? 1.The organizing committee of a winter carnival found that the profit P from the carnival depends on the ticket price t and the number of people who attend. The estimated profit is given by P =! 37 t t! Find the ticket price that will maximize the profit. Find the expected profit at this price. 68

39 Chapter 4. A ball is thrown straight down from a 180 m high cliff.. The relation h =! 5 t! 5t is a model that gives the approximate height of the ball h, in metres, at t seconds after it is thrown. How long does it take the ball to reach a ledge 80 m from the base of the cliff? Check the answer using a graphing calculator. 3. The population of a city is modelled by the relation P = 0.5 t + 10t + 00, where P is the population in thousands and t is the time in years. ( Note t = 0 corresponds to the year 000). a) what is the population in 000? b) what is the population in 00? c) when is the population 350,000? Explain your answer. d) use a graphing calculator to check the answers. 4. The population P of a city is modelled by P = 14 t + 80 t , where t is the time in years. When t = 0 the year is 000. a) what will the population be in 008? b) what was the population in 1991? c) in which year(s) was the population 30,000? d) determine the year when the fewest people lived in the city. 5. The expression A =! w + 36 w models the area of a field, where w is the width in metres and A is the area in square metres. What dimensions produce an area of 11 m? 6. A pair of skydivers jump out of an airplane 5.5 km above the ground. The equation H = 5500! 5 t is an approximate model for the divers altitude in metres at t seconds after jumping out of the plane. a) after 10 s how far have the divers fallen? b) they open their chutes at an altitude of 1000 m. How long did they fall? c) If the parachute does not open at 1000 m, how much time is left to use the emergenct chute? 7. A professional stunt performer at a theme park dives off a tower 1 m high into the water. The diver s height above the ground at time t seconds is given by the equation h =! 4.9 t + 1. a) how long does it take to reach the halfway mark? b) how long does it take to reach the water? c) compare the times in (a) and (b). Explain why the time at the bottom is not twice the time at the halfway point. 8. Water from a fire hose is sprayed on a fire 15 m up the side of a wall. The equation H =! x + x models the height of the jet of water and the horizontal distance from the nozzle in metres. a) what is the farthest distance back from the building that a firefighter could stand and still reach the fire? Explain. Include a diagram with your explanation. Quadratic Formula 69

40 9. A digital sensor records the height of a baseball after it is hit into the air. Quadratic regression on the data gives the quadratic relation y =! 4.9 x x How long is the ball in the air? Verify your answer using graphing technology. 30. A cliff diver in Acapulco, Mexico, dives from about 17 m above the water. The diver s height above the water h, in metres, after t seconds is modelled by h =! 4.9t t How long is the diver in the air? 70