Quadratic Functions Unit

Transcription

1 Quadratic Functions Unit (Level IV Academic Math) NSSAL (Draft) C. David Pilmer 009 (Last Updated: Dec, 011) Use our online math videos. YouTube: nsccalpmath

2 This resource is the intellectual property of the Adult Education Division of the Nova Scotia Department of Labour and Advanced Education. The following are permitted to use and reproduce this resource for classroom purposes. Nova Scotia instructors delivering the Nova Scotia Adult Learning Program Canadian public school teachers delivering public school curriculum Canadian nonprofit tuition-free adult basic education programs The following are not permitted to use or reproduce this resource without the written authorization of the Adult Education Division of the Nova Scotia Department of Labour and Advanced Education. Upgrading programs at post-secondary institutions Core programs at post-secondary institutions Public or private schools outside of Canada Basic adult education programs outside of Canada Individuals, not including teachers or instructors, are permitted to use this resource for their own learning. They are not permitted to make multiple copies of the resource for distribution. Nor are they permitted to use this resource under the direction of a teacher or instructor at a learning institution. Acknowledgments The Adult Education Division would like to thank the following university professors for reviewing this resource to ensure all mathematical concepts were presented correctly and in a manner that supported our learners. Dr. Genevieve Boulet (Mount Saint Vincent University) Dr. Robert Dawson (Saint Mary s University) The Adult Education Division would also like to thank the following NSCC instructors for piloting this resource and offering suggestions during its development. Charles Bailey (IT Campus) Elliott Churchill (Waterfront Campus) Barbara Gillis (Burridge Campus) Barbara Leck (Pictou Campus) Suzette Lowe (Lunenburg Campus) Floyd Porter (Strait Area Campus) Brian Rhodenizer (Kingstec Campus) Joan Ross (Annapolis Valley Campus) Jeff Vroom (Truro Campus)

3 Table of Contents Introduction.. i Negotiated Completion Date i The Big Picture ii Course Timelines. iii Introduction to Quadratic Functions 1 Using the Graphing Calculator to Interpret Quadratic Functions The Most Basic Quadratic Function 8 Quadratic Functions and Transformations... 9 State the Transformations: Quadratic Functions.. 1 Visualizing Quadratic Functions.. 15 Graphing Quadratic Functions Using Transformations Using Finite Differences to Identify Quadratic Functions... 6 Using Finite Differences to Determine the Equation... 1 Find the Equation Given Three Points. 0 Find the Equation Given the Verte and a Point. Putting It Together Part 1 50 Multiplying Polynomials. 5 Factoring Part 1 66 Factoring Part 7 Alternate Forms of Decomposition (Optional) 80 Factoring Part 85 Standard Form to Transformational Form Solving Quadratic Equations by Factoring.. 99 Solving Quadratic Equations Using the Quadratic Formula 105 Programming the Quadratic Formula into a TI-8 or TI-8 11 Word Problems Involving Quadratic Equations. 11 Finding the Verte Application Questions.. 1 Putting It Together Part. 19 Post-Unit Reflections Additional Practice: Multiplying Polynomials (Optional) Additional Practice: Factoring Polynomials (Optional) Additional Practice: Standard Form to Transformational Form (Optional) 15 Additional Practice: Word Problems Involving Quadratic Equations (Optional).. 15 Review Sheet: Types of Factoring Factoring Flow Chart Terminology. 158 Answers 159 NSSAL i Draft 009 C. D. Pilmer

4 Introduction In this unit we will learn about quadratic functions. The graphs of these functions form curves called parabolas. Their equations can be written in the transformation form y = k( c) d or in the standard form y = a b c. We will learn to work with both forms of the equations and see how this type of function can be used to model a variety of real world applications and be used to answer questions regarding those applications. It is important to note that different teachers and professors use different terminology to describe these two forms of the quadratic functions. Many refer to y = a b c as the general form and y = k( c) d as the standard form. Please be aware of this if you are using other resources to assist you with the concepts covered in this unit. This is a very lengthy unit because many smaller concepts have to be taught prior to learning the larger concepts directly connected to quadratic functions. For eample, before one can learn how to solve quadratic equations, one must know how to multiply polynomial epressions and factor polynomial epressions. Similarly, before one can learn how to graph quadratic functions, one must learn what transformations are and how they affect curves. The key to success in this unit is recognizing how all of the smaller concepts connect and give one a broad understanding of quadratic functions. In this unit learners will do the following. Interpret graphs of real world phenomena. All of the graphs will be quadratic functions. Graph quadratic functions of the form y = k( c) d using transformations. Determine the equation of a quadratic function in a variety of ways. Multiply and factor polynomial epressions. Change quadratic functions from their standard form to their transformational form. Solve quadratic equations by factoring and using the quadratic formula. Determine the coordinates of the verte of a quadratic function. Solve multi-step multi-concept application questions All of this will be accomplished while looking at a variety of real world applications of quadratic functions and revisiting concepts covered in prior units (e.g. domain, range, intercepts, ). Negotiated Completion Date After working for a few days on this unit, sit down with your instructor and negotiate a completion date for this unit. Start Date: Completion Date: Instructor Signature: Student Signature: NSSAL ii Draft 009 C. D. Pilmer

5 The Big Picture The following flow chart shows the optional bridging unit and the eight required units in Level IV Academic Math. These have been presented in a suggested order. Bridging Unit (Recommended) Solving Equations and Linear Functions Describing Relations Unit Relations, Functions, Domain, Range, Intercepts, Symmetry Systems of Equations Unit by Systems, Plane in -Space, by Systems Trigonometry Unit Pythagorean Theorem, Trigonometric Ratios, Law of Sines, Law of Cosines Sinusoidal Functions Unit Periodic Functions, Sinusoidal Functions, Graphing Using Transformations, Determining the Equation, Applications Quadratic Functions Unit Graphing using Transformations, Determining the Equation, Factoring, Solving Quadratic Equations, Verte Formula, Applications Rational Epressions and Radicals Unit Operations with and Simplification of Radicals and Rational Epressions Eponential Functions and Logarithms Unit Graphing using Transformations, Determining the Equation, Solving Eponential Equations, Laws of Logarithms, Solving Logarithmic Equations, Applications Inferential Statistics Unit Population, Sample, Standard Deviation, Normal Distribution, Central Limit Theorem, Confidence Intervals NSSAL iii Draft 009 C. D. Pilmer

6 Course Timelines Academic Level IV Math is a two credit course within the Adult Learning Program. As a two credit course, learners are epected to complete 00 hours of course material. Since most ALP math classes meet for 6 hours each week, the course should be completed within 5 weeks. The curriculum developers have worked diligently to ensure that the course can be completed within this time span. Below you will find a chart containing the unit names and suggested completion times. The hours listed are classroom hours. In an academic course, there is an epectation that some work will be completed outside of regular class time. Unit Name Minimum Completion Time in Hours Maimum Completion Time in Hours Bridging Unit (optional) 0 0 Describing Relations Unit 6 8 Systems of Equations Unit 18 Trigonometry Unit 18 0 Sinusoidal Functions Unit 0 Quadratic Functions Unit 6 Rational Epressions and Radicals Unit 1 16 Eponential Functions and Logarithms Unit 0 Inferential Statistics Unit 0 Total: 150 hours Total: 00 hours As one can see, this course covers numerous topics and for this reason may seem daunting. You can complete this course in a timely manner if you manage your time wisely, remain focused, and seek assistance from your instructor when needed. NSSAL iv Draft 009 C. D. Pilmer

7 Introduction to Quadratic Functions In this unit we will be looking at quadratic functions. Quadratic functions are of the form y = a b c (standard form) or can also be written in the form y = k( c) d (transformational form). We will learn about these two different forms at a later date. Note that in both cases the function uses the second power (square) as the highest power of the unknown. When quadratic functions are graphed, they form curves called parabolas. Rather than starting with all the algebra associated with quadratic functions, let s start with some real world applications. One of the most common applications is projectile motion. When a person throws a baseball to another person, the trajectory or path of the ball is referred to the motion of the projectile. As you will see, there are other applications of quadratic functions, many associated with business where one is attempting to minimize cost or maimize profit. Eample 1 A soccer ball is kicked from ground level. Its flight path is shown on the graph. The height of the ball, in metres, is on the vertical ais. The distance the ball travels horizontally, in metres, is on the horizontal ais. (a) What is the maimum height reached by the ball? (b) How far does the ball travel horizontally when it reaches this maimum height? (c) What s the initial height of the soccer ball? (d) How far will the ball travel horizontally before it hits the ground? (e) Approimate the height of the ball after it has traveled metres horizontally. (f) Determine the horizontal distances that correspond to a height of metres. (g) State the domain and range. Answers: (a) The ball reaches a maimum height of metres. (b) To reach its maimum height, the ball must travel 6 metres horizontally. (c) The initial height of the soccer ball is 0 metres. (d) The ball travels 1 metres horizontally before it hits the ground. (e) After the ball has traveled metres horizontally, it reaches an approimate height of. metres. (f) The ball is at a height of metres at two instances; as the ball goes up and as it comes down. This will occur when the horizontal distances are metres and 9 metres. R 0 d 1 hε R 0 h (g) Domain: { dε } Range: { } Horizontal Distance in Metres Note: The flight paths of projectiles are not truly parabolic when we factor in wind resistance. For our purposes, using a quadratic function to model these situations is acceptable as long as we recognize the limitations of this mathematical model. Height in Metres 5 1 NSSAL 1 Draft 009 C. D. Pilmer

8 Questions: 1. A baseball is thrown. Its flight path is shown on the graph. The height of the ball, in metres, is on the vertical ais. The distance the ball travels horizontally, in metres, is on the horizontal ais. Height in Metres Horizontal Distance in Metres (a) (b) (c) (d) (e) (f) (g) (h) Approimately how far will the ball travel horizontally before it hits the ground? Approimate the height of the ball after it has traveled 1 metre horizontally. What is the maimum height reached by the ball? How far does the ball travel horizontally when it reaches this maimum height? From what height was the baseball thrown? Approimate the horizontal distances that correspond to a height of metres. State the domain of this function. State the range of this function. Answers. Akira s company manufactures alternators for a large international automotive company. If her company manufactures too few alternators, she can not make a profit. If the large international company requests too many alternators, her profits drop because she has to pay overtime to her workers, and hire eternal trucking companies to deliver the additional parts. There is also a limit to the number of alternators her company is capable of producing. The following graph shows the relationship between the monthly profits her company can make and the number of alternators she produces in a month for the large international company. The profits are measure in tens of thousands of dollars. The number of alternators is measured in thousands. Profit Number of Alternators Produced NSSAL Draft 009 C. D. Pilmer

9 (a) (b) (c) (d) (e) (f) (g) How many alternators should Akira s company produce in a month to maimize her company s profit. If her company does not produce any alternators, what will her losses be for that month? If no profit is made in a month, how many alternators must be produced? If they produce 6000 alternators in a month, what will be the epected monthly profit? If the monthly profit is $ , how many alternators were produced? State the domain. State the range. Answers. The largest radio telescope in the world is the Arecibo telescope located in Puerto Rico. The reflecting surface for the telescope is embedded in the ground. If one was to cut the reflecting surface in half, the resulting curve would be formed. This graph shows the depth in feet of the reflecting surface relative to the distance in feet from the upper edge of the surface depth (ft) horizontal distance (ft) (a) (b) (c) (d) (e) (f) (g) Approimately how wide is the reflective surface on the telescope? Approimately how far down is the lowest point on the reflective surface? At what horizontal distances is the reflective surface 0 feet below ground level? State the depth intercept. State the domain. State the range. Is this function odd, even or neither? Answers NSSAL Draft 009 C. D. Pilmer

10 Using the Graphing Calculator to Interpret Quadratic Functions Eample 1 A garden hose sprays a stream of water across a lawn. The quadratic function h = 0.5d d 1 describes the height, h, of the stream of water above the lawn in terms of the horizontal distance, d, from the hose nozzle. Both the height and horizontal distance are measured in metres. Using graphing technology, answer each of these questions. Use the following WINDOW settings. (a) What is the maimum height reached by the water? (b) What horizontal distance corresponds to the maimum height of the stream of water? (c) What height is the hose nozzle? (d) At what horizontal distance will the water hit the ground? (e) If the water pressure was increased significantly, what feature on the graph would not change? Answer: (a) You first have to enter the function and graph it on the calculator. Y = > (Enter the function.) > WINDOW > (Adjust window settings.) > GRAPH Using the TRACE command and the right and left arrows, you can move to the highest point on the curve. Coordinates are typically of the form (,y) however, in this situation they are really (d,h). Since we want the maimum height for this question, we ll use the number 5. The water stream reaches a maimum height of 5 m. (b) You don t have to adjust anything on the calculator to do this part of the question. For this question you want the horizontal distance that corresponds to the maimum height. This will be the (or d) coordinate of the point you found in part (a). The horizontal distance corresponding to the maimum height is m. (c) Using the TRACE command, you find that the hose nozzle is at a height of 1 m. NSSAL Draft 009 C. D. Pilmer

11 (d) Using the TRACE command, you find that the water will hit the ground at approimately 8.5 m. (e) If the water pressure is increased the water stream would go higher and a further horizontal distance. The only thing that wouldn t change is the initial height of the water (i.e. the height of the hose nozzle). The y- (or h-) intercept would remain the same. Questions: 1. A baseball is thrown vertically into the air. It s height with respect to time can be described by the quadratic function h = 16t -.9t, where h is the height in metres and t is the time in seconds. (a) Graph the equation on a graphics calculator using the following Window settings. Sketch the graph in the space provided. Parts (b) to (l) are multiple choice questions. (b) The vertical ais (dependent variable) represents (i) the height of the ball off the ground. (ii) the speed of the ball. (iii) the time. (iv) the spot where the ball hits the ground. (v) the distance the ball travels horizontally. (c) The horizontal ais (independent variable) represents (i) the height of the ball off the ground. (ii) the speed of the ball. (iii) the time. (iv) the spot where the ball hits the ground. (v) the distance the ball travels horizontally. (d) The graph of this equation is a (i) straight line formed by a linear function. (ii) straight line formed by a quadratic function. (iii) parabola formed by a linear function. (iv) parabola formed by a quadratic function. (v) none of these NSSAL 5 Draft 009 C. D. Pilmer

12 (e) What is the initial height of the ball before it was thrown? (i) 0 m (ii) m (iii). m (iv) m (v) 6 m (f) Approimate the maimum height reached by the ball. (i) 0 m (ii) 1.6 m (iii) 10 m (iv) 15 m (v) 18 m (g) When does the ball reach its maimum height? (i) 0 s (ii) 1. s (iii) 1.6 s (iv) 15 s (v) 0 s (h) Approimate the height of the ball 1 second after it is thrown. (i) 1 m (ii) 1 m (iii) 1 m (iv) 15 m (v) 18 m (i) Approimate the time when the ball reaches a height of 10 m. (i) 0.6 s (ii).6 s (iii) 0.8 s (iv) both 0.6 s and.6 s (j) When does the ball strike the ground? (i) 1.6 s (ii).7 s (iii).1 s (iv). s (v).7 s (k) Determine the domain and range. R 0 t. (i) Domain: { tε } Range: { hε R 0 h 15 } (ii) Domain: { hε R 0 h 15 } Range: { tε R 0 t. } (iii) Domain: { tε R 0 t 1.6 } Range: { hε R h 15 } (iv) Domain: { hε R h 15 } Range: { tε R 0 t 1.6 } (l) Another individual throws a rock and the equation which describes its height with respect to time is h = 1t -.9t. How are these two situations (ball thrown/rock thrown) similar? (i) The ball and rock hit the ground at the same time. (ii) The ball and rock reached the same maimum height. (iii) The ball and rock were thrown from the same height. (iv) The ball and rock were thrown at the same speed. (v) none of these. The distance a car travels after the driver decides to slam on the brakes must consider two factors: the distance the car travels as the driver reacts to the situation (no brakes applied) and the distance the car travels when the brakes have been applied. The quadratic equation d = 0.007s 0. s describes the stopping distance, d, in terms of the initial speed, s, of the car. The distance is measured in metres and the speed is measured in kilometers per hour. Use graphing technology to answer the following questions. Use the following WINDOW settings. (Note: You will only be viewing half of the parabola.) NSSAL 6 Draft 009 C. D. Pilmer

13 (a) What is the y- (or d-) intercept and what does it represent in this situation? (b) If the stopping distance was 5 m, how fast was the driver going when he/she decided to slam on the brakes? (c) If the car was initially traveling at 100 km/h before the driver decides to hit the brakes, how far will the car travel before it comes to a stop? (d) If the car was initially traveling at 10 km/h before the driver decides to hit the brakes, how far will the car travel before it comes to a stop? (e) How do you feel about how the answers to (c) and (d) compare?. When a car is driven, the amount of gas consumed per kilometer changes based on the speed of the car. The quadratic equation c = s 0.116s 11.1 describes cost of gas per kilometre in terms of the speed of the vehicle. The cost is measured in cents per kilometre and the speed is measured in km/h. Use graphing technology to answer the following questions. Use the following WINDOW settings. (a) What speed is most cost-efficient? (b) At what speeds are you getting 9 cents per kilometre? (c) What is the cost per kilometre for a speed of 90 km/h? Think About Up to this point we have used graphs or graphing technology to understand situations that can be modeled using quadratic functions. Obviously there are many algebraic skills that we will have to learn so that we don t have to rely only on graphs and graphing technology. Over the net few weeks you will learn how to use paper and pencil techniques to: 1. graph quadratic functions. determine the equations of quadratic functions. solve quadratic equations associated with quadratic functions NSSAL 7 Draft 009 C. D. Pilmer

14 The Most Basic Quadratic Function The most basic quadratic function is y =. It has not undergone any transformations (i.e. reflections, stretches or translations). We will graph this function by generating a table of values using -values from - to, plotting the points, and connecting them with a smooth curve. y = y Please note that this type of curve is called a parabola. 1. State the domain.. State the range.. The verte of a parabola is either the highest or lowest point on the curve depending of the orientation of the curve. State the coordinates of the verte. Verte:. The ais of symmetry for a quadratic function is the vertical line that cuts the curve into two identical halves. State the equation of the ais of symmetry. Ais of Symmetry: 5. Is the function odd, even, or neither? 6. Why is this type of curve classified as a function? NSSAL 8 Draft 009 C. D. Pilmer

15 Quadratic Functions and Transformations In the last section we graphed the most basic quadratic function y = using a table of values. The resulting cupshaped curve is called a parabola. It has it s verte at ( 0,0) and the equation of the ais of symmetry is = 0. The curve is concave upwards. Name: Let s make this parabola and its corresponding table of values on a graphics calculator. Press Y =, enter the function ( y = ), and set the ZOOM feature to ZStandard. To generate the table of values, press TBLSET, set TblStart = - and set Tbl = 1. Now press TABLE. Complete the chart below. Our Basic Quadratic Function y = Enter: Y1=X^ Table of Values TblStart = - Tbl = 1 y Sketch of Graph Investigation: In each part of this investigation you are going to alter the equation of the quadratic function y = and see what effect this has on the graph and table of values. You will also identify the type of transformation (vertical stretch, horizontal stretch, vertical translation, horizontal translation or reflection in -ais) that has occurred. It is important to note that in each part of this investigation, you will be comparing your transformed quadratic function to the quadratic function y =. You will also have to determine the mapping rule. The mapping rule eplains how the ordered pairs in our original function, y =, have been changed to the NSSAL 9 Draft 009 C. D. Pilmer

16 ordered pairs in our new function. Graph each function and generate its table of values using technology. Note that the TblStart values change in parts (f) and (g) of this investigation. Complete the following chart. For the first few parts, the chart is partially completed. New Quadratic Table of Function Values (a) y = TblStart = - Tbl = 1 y Sketch of Graph Mapping Rule and Transformation (, y) (, y) We re dealing with a reflection in the -ais because the parabola is now upside down (i.e. concave downwards). (b) y = TblStart = - Tbl = 1 y (, y) (, y) (c) 1 y = TblStart = - Tbl = 1 y - (, y) (, ) We re dealing with a vertical stretch of 1 because the parabola appears wider. NSSAL 10 Draft 009 C. D. Pilmer

17 New Quadratic Table of Function Values (d) y = TblStart = - Tbl = 1 Sketch of Graph Mapping Rule and Transformation (, y) (, ) y (e) y = 5 TblStart = - Tbl = 1 (, y) (, ) y (f) y = ( ) TblStart = -6 Tbl = 1 (, y) (, ) y NSSAL 11 Draft 009 C. D. Pilmer

18 New Quadratic Function Table of Values (g) y = ( ) TblStart = 1 Tbl = 1 y 1 9 Sketch of Graph Mapping Rule and Transformation (, y) (, ) Summarize Your Findings: Equation Transformation (a) y = reflection in the -ais (b) y = (c) 1 1 y = vertical stretch of (d) y = (e) y = 5 (f) y = ( ) (g) y = ( ) Conclusions: If a quadratic function is of the form y = k( c) d, then: (i) the negative sign in front of the k indicates that a has occurred. (ii) the k indicates that a has occurred. (iii) the d indicates that a has occurred. (iv) the c indicates that a has occurred. The form y = k( c) d is referred to as the transformational form of the equation. If k is negative, then we know that we are also dealing with a reflection in the -ais. NSSAL 1 Draft 009 C. D. Pilmer

19 Questions 1. What transformation affects the -coordinate of the verte?. What transformation affects the y-coordinate of the verte?. Which transformation determines whether the parabola is concave upwards or concave downwards? Eamples For each of the following, state the transformations, state the coordinates of the verte, and determine whether the parabola is concave upwards or concave downwards. (a) y = 6( 7) 1 1 (b) y = ( 9) Answers: (a) y = 6( 7) 1 Reflection in the -ais, VS = 6, VT = 1, HT = 7 The verte is found by looking at the horizontal translation and vertical translation. The coordinates of the verte are (7, 1). The parabola is concave downwards because the quadratic function has undergone a reflection in the -ais. 1 y = 9 (b) ( ) VS = 1, HT = -9 The verte is found by looking at the horizontal translation and vertical translation. The coordinates of the verte are (-9, 0). The parabola is concave upwards because the quadratic function has not undergone a reflection in the -ais. NSSAL 1 Draft 009 C. D. Pilmer

20 State the Transformations: Quadratic Functions 1. In the previous section, you learned how to identify the transformations that occurred to y = when a quadratic function is presented in the form y = k( c) d. In the questions that follow, you will be given the equation of a quadratic function and be asked identify the transformations. Two eamples have been completed for you. Function e.g. ( ) Horizontal Translation Vertical Translation Reflection in the -ais Vertical Stretch y = none no e.g. ( ) ( ) 7 h = yes none (a) y = ( 7) 6 (b) y = g ( ) = 8 (c) y = ( ) (d) ( ) 5 (e) ( ) y = 7 (f) 5( ) 1 y = (g) ( ) 6( ) 1 h = (h) f ( ) = 8. Fill in the blanks. You may choose from the following terms. (Vertical Stretch, Vertical Translation, Horizontal Translation, Reflection in the -ais) (a) A will change a parabola from right side up (i.e. concave upwards) to upside down (i.e. concave downwards). (b) The -coordinate of the verte of a quadratic function can be determined by looking at the. (c) The y-coordinate of the verte of a quadratic function can be determined by looking at the. (d) The determines whether the graph is wider or narrower than the graph of y =. NSSAL 1 Draft 009 C. D. Pilmer

21 Visualizing Quadratic Functions We know that the graph of the quadratic function y = forms a cupshaped curve called a parabola. This function can be graphed on a graphing calculator. The resulting curve in ZStandard mode is displayed on the right. Most quadratic functions have undergone one or more transformations. The power of transformations is that we can visualize the resulting graph without having to do any formal work. For eample, if we have the quadratic function y = ( 5), we know the following. The parabola is concave downwards, due to the reflection in the -ais. The verte is located at (-5, ) due to the horizontal and vertical translations. The shape of the curve is narrower than that of y = due to the vertical stretch of. If you were asked to match this equation to one of the four graphs shown below, you would choose Graph B because it has all three features listed above. Graph A Graph B Graph C Graph D Correct Match Questions: 1. Match the equation to the appropriate graph. The scales on both the -ais and y-ais go from -10 to 10 (ZStandard mode). (a) y = ( ) Graph (i) Graph (ii) Graph (iii) Graph (iv) (b) y = ( 6) Graph (i) Graph (ii) Graph (iii) Graph (iv) NSSAL 15 Draft 009 C. D. Pilmer

22 (c) y = ( ) Graph (i) Graph (ii) Graph (iii) Graph (iv) (d) y = ( 5) Graph (i) Graph (ii) Graph (iii) Graph (iv) 1 (e) y = Graph (i) Graph (ii) Graph (iii) Graph (iv) (f) y = 1 ( ) Graph (i) Graph (ii) Graph (iii) Graph (iv) (g) y = ( ) 9 Graph (i) Graph (ii) Graph (iii) Graph (iv) NSSAL 16 Draft 009 C. D. Pilmer

23 (h) y = Graph (i) Graph (ii) Graph (iii) Graph (iv) (i) y = ( 6) Graph (i) Graph (ii) Graph (iii) Graph (iv) 1 (j) y = ( ) 5 Graph (i) Graph (ii) Graph (iii) Graph (iv). For each of the functions below, answer the following. Determine the coordinates of the verte. Describe the concavity (upwards or downwards) Describe the shape of the curve compared to that for the function narrower, wider, or no change. Two sample questions have been completed. y =. Use the terms Function Verte Concavity Shape e.g. ( ) y = (, 0) upwards narrower e.g. ( ) ( ) 7 f = 5 (-5, 7) downwards no change (a) y = ( ) 6 1 (b) y = ( ) 9 (c) y = 8 10 (d) g ( ) = ( ) 5 NSSAL 17 Draft 009 C. D. Pilmer

24 Graphing Quadratic Functions Using Transformations Before you can graph any quadratic function using transformation, you must understand what the table of values and graph for the function y = look like. The function y = is the most basic quadratic function; it hasn t undergone any transformations. y = y Notice the following for the graph. The verte is at (0, 0). The curve is concave upwards. Domain { ε R} yε R y 0 Range { } Using transformations, mapping rules, and your understanding of the basic quadratic function y =, you can graph any quadratic function of the form y = k( c) d. This can be accomplished using the following procedure. 1. State the transformations. Separate the transformations that affect the -values (horizontal translations) from those that affect the y-values (reflections in the -ais, vertical stretches, and vertical translations).. Construct the mapping rule.. Using the mapping rule, create the table of values for the desired quadratic function by altering the -values and y-values for y =.. Graph the points and draw the curve. 5. Check to see if the graph looks reasonable. Is it parabolic in shape? Does the location of the verte correspond to the horizontal and vertical translations? If the function has undergone a reflection in the -ais, is the curve concave downwards? If the function has undergone a vertical sketch, does the graph look correspondingly narrower or wider? Eample 1: Graph the function = ( ) 8 y using transformations. Answer: - Horizontal Translation of - Reflection in the -ais - Vertical Stretch of - Vertical Translation of 8 NSSAL 18 Draft 009 C. D. Pilmer

25 The mapping rule is created by eamining the transformations that affect the -values in the table, and eamining the transformations that affect the y-values. The -values are only affected by horizontal translations. In this case, the function has undergone a horizontal translation of such that the -values increase by. That is why the mapping rule shows that the -values change to. The y-values are affected by reflections in the -ais, vertical stretches, and vertical translations. In this case, the function has undergone a reflection in the -ais, a vertical stretch of, and a vertical translation of 8. That is why the mapping rule shows that the y-values change to y 8., y, y 8 Mapping Rule: ( ) ( ) Old Table ( y = ) New Table ( y = ( ) 8) y y - 9 ( ) = = 9 - ( ) = = -1 1 ( ) = = ( ) = = ( ) = = 1 ( ) = = = 9 9 ( ) = 0-10 ( ) ( ) ( ) ( ) ( ) ( ) ( ) The graph looks reasonable for the following reasons. The graph is concave downwards due to the reflection in the -ais. The coordinates of the verte, (, 8), correspond to the horizontal and vertical translations. The graph looks narrower than the graph of y = due to the vertical stretch of. Eample : Graph the function ( ) 1 y = using transformations. Answer: 1 - Horizontal Translation of - - Vertical Stretch of or 0.5 NSSAL 19 Draft 009 C. D. Pilmer

26 Mapping Rule: (, y) (, 0.5y) or (, y), y y y - 9 ( ) = -7.5 = 0.5( 9) - ( ) = -6 = 0.5( ) -1 1 ( 1 ) = = 0.5( 1) 0 0 ( 0 ) = - 0 = 0.5( 0) 1 1 ( 1 ) = = 0.5( 1) ( ) = - = 0.5( ) 9 ( ) = -1.5 = 0.5( 9) The graph looks reasonable for the following reasons. The graph is concave upwards because we do not have a reflection in the -ais. The coordinates of the verte, (-, 0), correspond to the horizontal and vertical translations. The graph looks wider than the graph of y = due to the vertical stretch of 0.5. Eample : Graph the function ( 5) 1 y = using transformations. Answer: - Horizontal Translation of 5 - Reflection in the -ais - Vertical Translation of -1 Mapping Rule: (, y) ( 5, y 1) y y ( ) 5 = 1 1 ( ) 5 = ( ) 5 = - 9 ( ) = -10 = ( 9) 1 - ( ) = -5 = ( ) ( 1 ) = - = ( 1) = ( 0) = ( 1) = ( ) 1 9 ( ) 5 = 8-10 = ( 9) 1 NSSAL 0 Draft 009 C. D. Pilmer

27 The graph looks reasonable for the following reasons. The graph is concave downwards because we have a reflection in the -ais. The coordinates of the verte, (5, -1), correspond to the horizontal and vertical translations. The graph does not look wider or narrower than the graph of y = because we do not have a vertical stretch. Questions: 1. For each quadratic function, state the transformations, and construct the mapping rule. The first question is partially completed. Please note that in the first question, the transformations that affect the -values are on the left opposed to the transformations that affect the y-values which are found on the right. If we do this, people generally find it easier to construct the mapping rule. (a) y = 5( 7) 1 Transformations: - Horizontal Translation of 7 - Reflection in the -ais Mapping Rule: (, y) ( 7, ) - Vertical Stretch of - Vertical Translation of (b) ( 8) 5 y = Transformations: Mapping Rule: (, y) (, ) NSSAL 1 Draft 009 C. D. Pilmer

28 1 y = 5 Transformations: (c) ( ) Mapping Rule: (, y) (, ) 1 (d) y = 1 5 Transformations: Mapping Rule: (, y) (, ). This question is partially completed. Complete the question. Graph ( ) 9 y = using transformations. - Horizontal Translation of - - Vertical Stretch of - Vertical Translation of -9 Mapping Rule (, y) (, y 9) y y - 9 ( ) = - ( ) = -1 1 ( ) = = ( 9) 9 = ( ) 9 1 = ( 1) The graph looks reasonable for the following reasons. The graph is concave because we do not have a reflection in the -ais. The coordinates of the verte, (, ), correspond to the horizontal and vertical translations. The graph looks than the graph of y = because we have a vertical stretch of. NSSAL Draft 009 C. D. Pilmer

29 . This question is partially completed. Complete the question. y = 1 using transformations. Graph ( ) - Horizontal Translation of - Reflection in the -ais - Vertical Stretch of 1 Mapping Rule (, y) (, ) y y ( ) = - ( ) = Vertical Translation of The graph looks reasonable for the following reasons. The graph is concave because we have a reflection in the -ais. The coordinates of the verte, (, ), correspond to the horizontal and vertical translations. The graph looks than the graph of y = because we have a vertical stretch of.. Graph each of the following quadratic functions using transformations. (a) ( ) 5 y = Transformations: Mapping Rule: y NSSAL Draft 009 C. D. Pilmer

30 (b) y = 10 Transformations: Mapping Rule: y Transformations: (c) y = ( ) 1 Mapping Rule: y For each of these quadratic functions, determine the coordinates of the verte whether the function is concave upwards or concave downwards whether the function is narrower, wider or the same as the function y =. (a) y = ( 7) 1 (b) y = ( 5) 10 (c) ( ) y = 1 (d) y = 11 Verte Concavity Narrower/Wider/Same NSSAL Draft 009 C. D. Pilmer

31 Verte Concavity Narrower/Wider/Same 1 g = 9 7 h = (e) y = ( 8) 6 (f) ( ) ( ) (g) ( ) 8 y = 1 using transformations. State the domain, range, coordinates of the verte and the equation of the ais of symmetry. 6. Graph ( 6) The height, h, of a projectile relative to the distance, d, it travels horizontally can be described by the equation h = 1 ( d ) 5. (a) Graph the function using transformations. (b) Use the graph to determine the maimum height reached by the projectile. (c) Determine the distance the projectile will travel horizontally when it reaches its maimum height. (d) Determine the initial height of the projectile. (e) State the equation of the ais of symmetry NSSAL 5 Draft 009 C. D. Pilmer

32 Using Finite Differences to Identify Quadratic Functions Previously we learned how to determine if a table of values could be represented by a linear function (i.e. y = m b). We discovered that if the -values were changing by the same increment and a common difference occurred between successive y-values, then the table of values was generated from a linear function. Eample 1: Determine if the following table of values can be modeled using a linear function. y Answer: y - 5 y In this particular eample, the -values are changing by increments of and the successive y- values display a common difference of. This table of values was generated by a linear function, specifically y = 8. If we can identify linear functions by looking for patterns in the table of values, can we do the same for quadratic functions? The answer is yes. Linear functions display a common difference at the D1 Level. Quadratic functions display a common difference at the D level. Consider the following table of values that was generated by the quadratic function y =. y y D ( 1) ( ) = D ( 1) = 6 6 = = = = = = = NSSAL 6 Draft 009 C. D. Pilmer

33 To the left we ve rewritten the table and started eamining it for patterns. As the -values change by the same increment (1), the y-values do not display a common difference. Since there is no common difference at the D1 level, we know that this table of values is not generated by a linear function. In the last column, however, when we take the differences of the differences, we do see a common difference. Common differences for quadratic functions occur at the D level. Check another quadratic function to see if this pattern holds up. Here is the table of values for the quadratic function y = 5. Analyze the table and determine if a common difference occurs at the D level? y y D D So how do we know that this common difference occurs at the D level for all quadratic functions? Consider the analysis of the function y = a b c; this equation can be used to represent all quadratic functions. A table of values using -values from 1 to 5 has been generated for this function. y = a b c 1 a b c a b c 9a b c 16a b c 5 5a 5b c Now we can analyze this table using finite differences. y = a b c 1 a b c D1 1 a b c a b D 1 9a b c 5a b a 1 16a b c 7a b a 1 5 5a 5b c 9a b a Notice that as the -values changed by the same increment (1), a common difference of a occurred at the D level. Using the general equation y = a b c, we have shown that all tables of values for quadratic functions will have a common difference at the D level when the -values are changing by the same increment. NSSAL 7 Draft 009 C. D. Pilmer

34 Eample : Determine if the following table of values can be modeled using a linear function, a quadratic function or neither. (a) (b) (c) y y y Answer: (a) y - - D ( 8) ( ) = D 1 ( 8) = = - -1 ( 1) = -1 ( 1) 10 = ( 50) ( 1) = -8 ( 8) ( 1) = ( 6) ( 8) = ( ) ( ) = Since the -values change by the same increment () and there is a common difference of - at the D level, then this table of values can be modeled using a quadratic function. (b) y D D Although there is a common difference at the D level, the -values are not changing by the same increment. This table does not appear to be generated by a linear or quadratic function. (c) y D D 1 9 NA NA NA NA NSSAL 8 Draft 009 C. D. Pilmer

35 Since the -values change by the same increment () and there is a common difference of 9 at the D1 level, then we know that this table of values can be modeled using a linear function. Questions: 1. Generate the table of values for y = 7 using -values from -1 to, and determine the common difference at the D1 level. y -1 D Generate the table of values for y = 1 using -values from - to, and determine the common difference at the D level. y - D1-1 D 0 1. Determine if the following table of values can be modeled using a linear function, a quadratic function or neither. (a) (b) y y 1 D1 1-1 D D 9 D NSSAL 9 Draft 009 C. D. Pilmer

36 (c) (d) y y - 5 D1 1 9 D1-8 D 7 D (e) (f) y y - 16 D D D - -0 D (g) (h) y y - -1 D D D.5 D NSSAL 0 Draft 009 C. D. Pilmer

37 Using Finite Differences to Determine the Equation We can identify quadratic functions using finite differences, but we can also use this technique to determine the equation of the quadratic function. The equation that we will generate will be in the form y = a b c. This is called the standard form of the equation. The procedure involves comparing the table of values for the function y = a b c to the table of values for our unknown quadratic function. Eample 1: Determine the equation of the function that is represented by the following table of values. y Answer: Start by analyzing the table using finite differences to see whether we are dealing with a quadratic function. y 1 9 D D We are dealing with a quadratic function because we have a common difference at the D level. Now we will generate the table of values for y = a b c using the same -values (1,,,, 5, 6). y = a b c 1 a b c D1 1 a b c a b D 1 9a b c 5a b a 1 16a b c 7a b a 1 5 5a 5b c 9a b a 1 6 6a 6b c 11a b a We can now compare the tables. Specific elements of one table are equal to specific elements in the other table. For eample, the a b c in the second table is equal to the 9 in the first table. Knowing these relationships we can generate the following three equations. a = a b = 6 a b c = 9 NSSAL 1 Draft 009 C. D. Pilmer

38 Now we can use these three equations to solve for a, b, and c. a = a = a =1 a b = 6 ( 1) b = 6 b = 6 b = a b c = 9 ( 1) ( ) c = 9 c = 9 c = 5 The equation of the quadratic function is y = 1 5 or y = 5. Eample : Determine the equation of the function that is represented by the following table of values. y Answer: Start by analyzing the table using finite differences to see whether we are dealing with a quadratic function. y -1 1 D D We are dealing with a quadratic function because we have a common difference at the D level. Now we will generate the table of values for y = a b c using the same -values (-1, 1,, 5, 7). y = a b c -1 a b c D1 1 a b c b D 9a b c 8a b 8a 5 5a 5b c 16a b 8a 7 9a 7b c a b 8a We can now compare the tables. Specific elements of one table are equal to specific elements in the other table. Knowing this we can generate the following three equations. 8a = 16 b = 8 a b c =1 NSSAL Draft 009 C. D. Pilmer

39 Now we can use these three equations to solve for a, b, and c. 8a = 16 b = 8 a b c =1 8a 8 = 16 b 8 = 8 ( ) ( ) c =1 6 c =1 a = b = c = 7 The equation of the quadratic function is y = 7. Eample : Determine the equation of the function that is represented by the following table of values. y Answer: y - - D1 0 0 D y = a b c - 9 a b c D1 0 c 9 a b D 9a b c 9 a b 18 a 6 6 a 6b c 7 a b 18 a 9 81 a 9b c 5 a b 18 a 18a = 18 18a 18 = a = 1 9a b = 9( 1) b = 9 b = b 15 = b = 5 9a b c = 9 ( 1) ( 5) c = 9 15 c = c = 9 15 c = 0 The equation of the quadratic function is y = or y = 5. NSSAL Draft 009 C. D. Pilmer

40 Eample : The main support cables on a particular suspension bridge form a parabolic curve when viewed from the side. The cables are suspended from two support towers 0 metres apart. The following table shows the height in metres of the cable above the roadway relative to the distance in metres from the vertical support tower on the left hand side of the bridge. (a) Determine the equation that describes the height, h, of the main support cables with respect to distance, d, from the left support tower. (b) Use the equation to determine the height of the main support cables when we are 7 metres from the left support tower. (c) Use the equation and graphing technology to determine the minimum height of the cable. (d) State the domain and range. Answers: (a) d h d 0 8 D D d h = ad bd c d 0 c D1 a b c a b D 16 a b c 1 a b 8 a 6 6 a 6b c 0 a b 8 a 8 6 a 8b c 8 a b 8 a Horizontal Distance (in metres) Height (in metres) a = 0.1 8a 0.1 = 8 8 a = a b = 1.1 c = 8 ( 0.015) b = b = 1.1 Therefore: b 1.0 = h = 0.015d 0.6d 8 b = 0.6 (b) h = 0.015d h = d 8 ( 7) 0.6( 7) h = h = The main support cable is 5.15 m above the roadway. NSSAL Draft 009 C. D. Pilmer

41 (c) Use the TRACE feature on a graphing calculator. To set the WINDOW setting, look at the information supplied in the question and the table of values. The minimum height of the main support cables is metres. (d) Domain: { dε R 0 d 0} Range: { hε R h 8} Note: Suspended cables form curves called hyperbolas; however, when subjected to a uniform load, as is the case with a suspension bridge, the cables deform and approach the shape of a parabola. Using a quadratic function to model the shape of these cables is acceptable, although not perfect. Questions: 1. Determine the equation of the function that is represented by the following table of values. The question has been partially completed. y Your Answer: y 1 - D D y = a b c 1 a b c D1 1 a b c a b D 1 9a b c 5 NSSAL 5 Draft 009 C. D. Pilmer

42 . Determine the equation of the function that is represented by the following table of values. y Determine the equation of the function that is represented by the following table of values. y NSSAL 6 Draft 009 C. D. Pilmer

43 . Sapphire and Manish are organizing a provincial softball tournament. It is a round-robin tournament where each team must play every other team eactly once. It s pretty easy to figure out how many games must be scheduled if only a few teams enter the tournament but what happens if many teams decide to participate. Sapphire and Manish want to see if they can use their knowledge of mathematics to address this issue. Number of Teams Number of Games Played (a) Determine the equation that describes the number of games, g, played in terms of the number of teams, t, signed up for the tournament. (b) Using the equation, determine number of games that must be played if 9 teams participate in the tournament. (c) Using the equation and graphing technology, determine how many teams signed up for the tournament if 66 games are played. (Hint: Remember to adjust the calculator s WINDOW settings. Please note that the -values on the calculator represent the number of teams, and the y-values represent the number of games played. Think about this when deciding upon Xmin, Xma, Ymin, and Yma values for the WINDOW settings. You will likely have to eperiment a little with these values until you find ones that give a good view of the function.) NSSAL 7 Draft 009 C. D. Pilmer

44 5. A projectile is fired vertically into the air. Eventually the projectile will reach its maimum height and fall back to the ground. Its height in metres is recorded at specific times. The time is measured in seconds. The following data was collected. t h (a) Determine the equation of the function that describes the height of the projectile with respect to time. (b) Using your equation from (a) and a graphing calculator, determine each of the following. maimum height reached by the projectile the time when the projectile reached its maimum height the initial height of the projectile the time when the projectile strikes the ground (c) State the domain and range. NSSAL 8 Draft 009 C. D. Pilmer

45 6. In 1971, Apollo 1 astronaut, Alan Shepard, hit a golf ball while on the surface of the moon. He made this shot using only one hand and while encumbered in a spacesuit. Although the eact trajectory of the ball was not recorded, we have taken liberal license and created our own data. Horizontal Distance Traveled (in metres) Height (in metres) (a) Determine the equation of the function that describes the height, h, of the golf ball with respect to horizontal distance, d, traveled. (b) Using the equation, determine the height of the ball after it traveled horizontally 10 m. (c) Using your equation and a graphing calculator, determine each of the following. maimum height reached by the golf ball the position where the ball strikes the moon s surface (d) State the domain and range. NSSAL 9 Draft 009 C. D. Pilmer