7.2 Properties of Graphs

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1 7. Properties of Graphs of Quadratic Functions GOAL Identif the characteristics of graphs of quadratic functions, and use the graphs to solve problems. LEARN ABOUT the Math Nicolina plas on her school s volleball team. At a recent match, her Nonno, Marko, took some time-lapse photographs while she warmed up. He set his camera to take pictures ever.5 s. He started his camera at the moment the ball left her arms during a bump and stopped the camera at the moment that the ball hit the floor. Marko wanted to capture a photo of the ball at its greatest height. However, after looking at the photographs, he could not be sure that he had done so. He decided to place the information from his photographs in a table of values. From his photographs, Marko observed that Nicolina struck the ball at a height of ft above the ground. He also observed that it took about 1.5 s for the ball to reach the same height on the wa down. Time (s) Height (ft) YOU WILL NEED ruler graph paper EXPLORE Parabolic skis are marketed as performing better than traditional straight-edge skis. Parabolic skis are narrower in the middle than on the ends. Design one side of a parabolic ski on a coordinate grid. In groups, discuss where an lines of smmetr occur and how the parabolic shape works in our design.? When did the volleball reach its greatest height? eample 1 Using smmetr to estimate the coordinates of the verte Marko s Solution Height (ft) Time (s) I plotted the points from m table, and then I sketched a graph that passed through all the points. The graph looked like a parabola, so I concluded that the relation is probabl quadratic. NEL 7. Properties of Graphs of Quadratic Functions 361

2 verte The point at which the quadratic function reaches its maimum or minimum value. Height (ft) Time (s) I knew that I could draw horizontal lines that would intersect the parabola at two points, ecept at the verte, where a horizontal line would intersect the parabola at onl one point. Using a ruler, I drew horizontal lines and estimated that the coordinates of the verte are around (.6, 8.). ais of smmetr A line that separates a -D figure into two identical parts. For eample, a parabola has a vertical ais of smmetr passing through its verte. Equation of the ais of smmetr: (.65, 8.) Time (s) Height (ft) From the equation, the -coordinate of the verte is.65. From the graph, the -coordinate of the verte is close to 8.. Therefore,.65 s after the volleball was struck, it reached its maimum height of approimatel 8 ft in. This means that the ball reached maimum height at just over 8 ft, about.6 s after it was launched. I used points that have the same -value, (, ) and (1.5, ), to determine the equation of the ais of smmetr. I knew that the ais of smmetr must be the same distance from each of these points. I revised m estimate of the coordinates of the verte. Reflecting A. How could Marko conclude that the graph was a quadratic function? B. If a horizontal line intersects a parabola at two points, can one of the points be the verte? Eplain. C. Eplain how Marko was able to use smmetr to determine the time at which the volleball reached its maimum height. 36 Chapter 7 Quadratic Functions and Equations NEL

3 APPLY the Math eample Reasoning about the maimum value of a quadratic function Some children are plaing at the local splash pad. The water jets spra water from ground level. The path of water from one of these jets forms an arch that can be defined b the function f () where represents the horizontal distance from the opening in the ground in feet and f () is the height of the spraed water, also measured in feet. What is the maimum height of the arch of water, and how far from the opening in the ground can the water reach? Manuel s Solution f () f () 5 f (1) 5.1(1) 1 3(1) f (1) f (1) f () I knew that the degree of the function is, so the function is quadratic. The arch must be a parabola. I also knew that the coefficient of, a, is negative, so the parabola opens down. This means that the function has a maimum value, associated with the -coordinate of the verte. I started to create a table of values b determining the -intercept. I knew that the constant, zero, is the -coordinate of the -intercept. This confirms that the stream of water shoots from ground level. I continued to increase b intervals of 1 until I noticed a repeat in m values. A height of 18.7 ft occurs at horizontal distances of 1 ft and 13 ft. Based on smmetr and the table of values, the maimum value of f () will occur halfwa between (1, 18.7) and (13, 18.7). The arch of water will reach a maimum height between 1 ft and 13 ft from the opening in the ground. maimum value The greatest value of the dependent variable in a relation. NEL 7. Properties of Graphs of Quadratic Functions 363

4 Height (ft) (1.5, 18.7) Horizontal distance (ft) I used m table of values to sketch the graph. I etended the graph to the -ais. I knew that m sketch represented onl part of the function, since I am onl looking at the water when it is above the ground. I used two points with the same -value, (1, 18.7) and (13, 18.7), to determine the equation of the ais of smmetr. Equation of the ais of smmetr: Height at the verte: f () f (1.5) 5.1(1.5) 1 3(1.5) f (1.5) 5.1(156.5) f (1.5) f (1.5) The water reaches a maimum height of ft when it is 1.5 ft from the opening in the ground. The water can reach a maimum horizontal distance of 5 ft from the opening in the ground. I knew that the -coordinate of the verte is 1.5, so I substituted 1.5 into the equation to determine the height of the water at this horizontal distance. Due to smmetr, the opening in the ground must be the same horizontal distance from the ais of smmetr as the point on the ground where the water lands. I simpl multiplied the horizontal distance to the ais of smmetr b. The domain of this function is # # 5, where [ R. Your Turn Another water arch at the splash pad is defined b the following quadratic function: f () a) Graph the function, and state its domain for this contet. b) State the range for this contet. c) Eplain wh the original function describes the path of the water being spraed, whereas the function in Eample 1 does not describe the path of the volleball. 36 Chapter 7 Quadratic Functions and Equations NEL

5 eample 3 Graphing a quadratic function using a table of values Sketch the graph of the function: 5 1 Determine the -intercept, an -intercepts, the equation of the ais of smmetr, the coordinates of the verte, and the domain and range of the function. Anthon s Solution 5 1 The function is a quadratic function in the form 5 a 1 b 1 c a 5 1 b 5 1 c The degree of the given equation is, so the graph will be a parabola. Since the coefficient of is positive, the parabola opens up. Since the -coordinate of the -intercept is less than zero and the parabola opens up, there must be two -intercepts and a minimum value. I made a table of values. I included the -intercept, (, ), and determined some other points b substituting values of into the equation. I stopped determining points after I had identified both -intercepts, because I knew that I had enough information to sketch an accurate graph I graphed each coordinate pair and then drew a parabola that passed through all the points. Equation of the ais of smmetr: I used the -intercepts to determine the equation of the ais of smmetr. minimum value The least value of the dependent variable in a relation. NEL 7. Properties of Graphs of Quadratic Functions 365

6 -coordinate of the verte: 5 (.5) 1 (.5) The verte is (.5,.5). The -intercept is. The -intercepts are and 1. The equation of the ais of smmetr is 5.5 The verte is (.5,.5). Domain and range: {(, ) [ R, $.5, [ R} I knew that the verte is a point on the ais of smmetr. The -coordinate of the verte must be.5. To determine the -coordinate of the verte, I substituted.5 for in the given equation. The verte, (.5,.5), defines the minimum value of. No restrictions were given for, so the domain is all real numbers. Your Turn Eplain how ou could decide if the graph of the function has -intercepts. eample Locating a verte using technolog A skier s jump was recorded in frame-b-frame analsis and placed in one picture, as shown. The skier s coach used the picture to determine the quadratic function that relates the skier s height above the ground,, measured in metres, to the time,, in seconds that the skier was in the air: Graph the function. Then determine the skier s maimum height, to the nearest tenth of a metre, and state the range of the function for this contet. 366 Chapter 7 Quadratic Functions and Equations NEL

7 Isidro s Solution I entered the equation into m calculator. To make sure that the graph models the situation, I set up a table of values. The skier s jump will start being timed at s, and the skier will be in the air for onl a few seconds, so I set the table to start at an -value of zero and to increase in increments of.5. I decided to set the minimum height at m it doesn t make sense to etend the function below the -ais, because the skier cannot go below the ground. I checked the table and noticed that the greatest -value is onl m, and that is negative at 3.5 s. I used these values to set an appropriate viewing window for the graph. I graphed the function and used the calculator to locate the maimum value of the function. The skier achieved a maimum height of 1.5 m above the ground 1.5 s into the jump. The range of the function is { # # 1.5, [ R}. In this situation, the height of the skier varies between m and 1.5 m. Your Turn On the net da of training, the coach asked the skier to increase his speed before taking the same jump. At the end of the da, the coach analzed the results and determined the equation that models the skier s best jump: How much higher did the skier go on this jump? NEL 7. Properties of Graphs of Quadratic Functions 367

8 In Summar Ke Idea A parabola that is defined b the equation 5 a 1 b 1 c has the following characteristics: If the parabola opens down (a, ), the verte of the parabola is the point with the greatest -coordinate. The -coordinate of the verte is the maimum value of the function. If the parabola opens up (a. ), the verte of the parabola is the point with the least -coordinate. The -coordinate of the verte is the minimum value of the function. The parabola is smmetrical about a vertical line, the ais of smmetr, through its verte. a b c a a b c, a verte (maimum value) verte (minimum value) ais of smmetr ais of smmetr Need to Know For all quadratic functions, the domain is the set of real numbers, and the range is a subset of real numbers. When a problem can be modelled b a quadratic function, the domain and range of the function ma need to be restricted to values that have meaning in the contet of the problem CHECK Your Understanding 1. a) Determine the equation of the ais of smmetr for the parabola. b) Determine the coordinates of the verte of the parabola. c) State the domain and range of the function. 368 Chapter 7 Quadratic Functions and Equations NEL

9 . State the coordinates of the -intercept and two additional ordered pairs for each function. a) f () b) f () 5 3. For each function, identif the - and -intercepts, determine the equation of the ais of smmetr and the coordinates of the verte, and state the domain and range. a) 6 b) PRACTISING. For each function, identif the equation of the ais of smmetr, determine the coordinates of the verte, and state the domain and range. a) 6 c) b) d) NEL 7. Properties of Graphs of Quadratic Functions 369

10 5. Each parabola in question is defined b one of the functions below. a) f () c) f () b) f () d) f () Identif the function that defines each graph. Then verif the coordinates of the verte that ou determined in question. 6. State whether each parabola has a minimum or maimum value, and then determine this value. a) b) c) a) Complete the table of values shown for each of the following functions. i) ii) 5 3 b) Graph the points in our table of values. c) State the domain and range of the function. 8. a) Graph the functions 5 and 5. b) How are the graphs the same? How are the graphs different? c) Suppose that the graphs were modified so that the became the graphs of 5 1 and 5 1. Predict the verte of each function, and eplain our prediction. 9. For each of the following, both points, (, ), are located on the same parabola. Determine the equation of the ais of smmetr for each parabola. a) (, ) and (6, ) c) (6, ) and (, ) b) (1, 3) and (9, 3) d) (5, 1) and (3, 1) 1. A parabola has -intercepts 5 3 and 5 9. Determine the equation of the ais of smmetr for the parabola. 11. a) Graph each function. i) f () iii) f () ii) f () iv) f () b) Determine the equation of the ais of smmetr and the coordinates of the verte for each parabola. c) State the domain and range of each function. 37 Chapter 7 Quadratic Functions and Equations NEL

11 1. In southern Alberta, near Fort Macleod, ou will find the famous Head-Smashed-In Buffalo Jump. In a form of hunting, Blackfoot once herded buffalo and then stampeded the buffalo over the cliffs. If the height of a buffalo above the base of the cliff, f (), in metres, can be modelled b the function f () where is the time in seconds after the buffalo jumped, how long was the buffalo in the air, to the nearest hundredth of a second? 13. In the game of football, a team can score b kicking the ball over a bar and between two uprights. For a kick in a particular game, the height of the ball above the ground,, in metres, can be modelled b the function where is the time in seconds after the ball left the foot of the plaer. a) Determine the maimum height that this kick reached, to the nearest tenth of a metre. b) State an restrictions that the contet imposes on the domain and range of the function. c) How long was the ball in the air? 1. An annual fireworks festival, held near the seawall in downtown Vancouver, choreographs rocket launches to music. The height of one rocket, h(t), in metres over time, t, in seconds, is modelled b the function h(t) 5.9t 1 8t Determine the domain and range of the function that defines the height of this rocket, to the nearest tenth of a metre. 15. Melinda and Genevieve live in houses that are net to each other. Melinda lives in a two-store house, and Genevieve lives in a bungalow. The like to throw a tennis ball to each other through their open windows. The height of a tennis ball thrown from Melinda to Genevieve, f (), in feet, over time,, measured in seconds is modelled b the function f () What are the domain and range of this function if Genevieve catches the ball ft above the ground? Draw a diagram to support our answer. NEL 7. Properties of Graphs of Quadratic Functions 371

12 16. Sid knows that the points (1, 1) and (5, 1) lie on a parabola defined b the function f () a) Does f () have a maimum value or a minimum value? Eplain. b) Determine, in two different was, the coordinates of the verte of the parabola. Closing 17. a) Eplain the relationship that must eist between two points on a parabola if the -coordinates of the points can be used to determine the equation of the ais of smmetr for the parabola. b) How can the equation of the ais of smmetr be used to determine the coordinates of the verte of the parabola? Etending 18. Gamez Inc. makes handheld video game plaers. Last ear, accountants modelled the compan s profit using the equation P This ear, accountants used the equation P In both equations, P represents the profit, in hundreds of thousands of dollars, and represents the number of game plaers sold, in hundreds of thousands. If the same number of game plaers were sold in these ears, did Gamez Inc. s profit increase? Justif our answer. 19. A parabola has a range of { # 1.5, [ R} and a -intercept of 1. The ais of smmetr of the parabola includes point (3, 5). Write the equation that defines the parabola in standard form if a Chapter 7 Quadratic Functions and Equations NEL