2-6. _ k x and y = _ k. The Graph of. Vocabulary. Lesson

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1 Chapter 2 Lesson 2-6 BIG IDEA The Graph of = _ k and = _ k 2 The graph of the set of points (, ) satisfing = k_, with k constant, is a hperbola with the - and -aes as asmptotes; the graph of the set of points (, ) satisfing = k_ 2 looks somewhat similiar but is closer to the aes. Vocabular hperbola branches of a hperbola vertical asmptote horizontal asmptote discrete set inverse-square curve In previous lessons ou eplored the graphs of the equations of the direct-variation functions = k and = k 2. Activit 1 eplores the quite different graph of the equation of an inverse-variation function, = k _. Activit 1 MATERIALS graphing utilit Step 1 Graph the function = _ 10 in a standard window. The resulting graph has points onl in Quadrants I and III. Step 2 Trace along the graph starting from a positive value of and moving to the right. You can trace beond the window edge if ou want. It ma also help if ou zoom in on the graph for large values of. Describe what happens to as increases in value. Mental Math m 1 = 72. What is m 2 if a. 1 and 2 are supplementar? b. 1 and 2 are the two acute angles in a right triangle? c. 1 and 2 are vertical angles? d. 1 and 2 form a linear pair? Step 3 Now trace to the left starting from a negative value of. Describe what happens to as decreases in value. Step 4 Does the graph of = _ 10 ever intersect the -ais? If it does, give the coordinates of an point(s) of intersection. If it does not, eplain wh not. Step 5 Now trace closer and closer to = 0 from both the positive and negative directions. Does the graph of = _ 10 ever intersect the -ais? If it does, give the coordinates of an point(s) of intersection. If it does not, eplain wh not. Step 6 Graph = k_ for a few nonzero values of k other than 10. Trace along the graphs past the edges of the standard window and near = 0. Do all the graphs have the same behavior as gets ver far from 0 and ver close to 0? 106 Variation and Graphs

2 The Graph of = k_ The graph of ever function with an equation of the form = k _, where k 0, is a hperbola. In Activit 1 ou graphed the hperbola = _ 10. All hperbolas with equation = _ k share some properties. For eample, the functions the represent all have the same domain and range. Eample 1 What are the domain and range of the function with equation = k_? Solution To fi nd the domain, think: What values can have? You can substitute an number for ecept 0 into the equation = k_. So, the domain of the function with equation = k_ is { 0}. To fi nd the range, think: What values can have? Recall from the Activit that can be positive or negative, large or small. But if = 0, ou would have 0 = k_, or 0 = k, which is impossible because, in the formula = k_ for a hperbola, k cannot be zero. So, the range of the function with equation = k_ is { 0}. Lesson 2-6 READING MATH Hperbola (a curve) and hperbole (an eaggeration) come from the same Greek word hperbole meaning ecess. Like parabola, hperbola refers to a comparison of two distances; however, one of these distances is in ecess of the other. The graphs in the = k _ famil of curves share another propert. Each hperbola in this famil consists of two separate parts, called branches. When k > 0, the branches of = k _ lie in the first and third quadrants; if k < 0, the branches lie in the second and fourth quadrants, as shown below. II I II I III IV III IV = k_ when k > 0 Domain { 0 }, Range { 0 } = k_ when k < 0 Domain { 0 }, Range { 0 } Asmptotes When = 0, = k _ is undefined, so, the curve does not cross the -ais. However, when is near 0, the function is defined. The Graphs of = k_ and = k_ 2 107

3 Chapter 2 You found in Activit 1 that for = k _, k > 0, as approaches zero from the positive direction, the -value gets larger and larger. Similarl, as approaches zero from the negative direction, the -value gets smaller and smaller (more and more negative). As the values of get closer and closer to a certain value a, if the values of the function get larger and larger or smaller and smaller, then the vertical line with equation = a is called a vertical asmptote of the graph of the function. The -ais is a vertical asmptote to the graph of = k _, for k 0. Similarl, the -ais is a horizontal asmptote of the curve = _ k. As gets ver, ver large (or ver, ver small), the value of gets closer and closer to zero. In some situations, onl one branch of a hperbola is relevant. For instance, recall the Condo Care Compan eample of Lesson 2-2. The time t it takes to complete the job and the number of workers w are related b the equation t = _ 48 w. A table of values for this equation is shown below, along with a graph of the points in the table at the right. Because ou cannot have a negative number of workers, there are no points in Quadrant III. w t _ _ _ 3 4 4_ _ 5 In this eample the function has a discrete domain, the set of positive integers. A discrete set is one in which there is a positive distance greater than some fied amount between an two elements of the set. Because the number of workers is an integer, it does not make sense to connect the points of this graph. Thus, the graph consists of a set of discrete points on one branch of the hperbola = _ 48. QY Activit 2 eplores the graphs of a different famil of inverse-variation functions, those with equation = _ k. We call the graph of such an 2 inverse-variation function an inverse-square curve. Activit 2 Step 1 Graph the function = _ 10 in a standard window. The resulting graph 2 appears in Quadrants I and II. Wh do ou think this graph does not appear in Quadrants III or IV? Step 2 Trace along the curve to a positive value of. Continue tracing to the right. Describe what happens to as increases. 2 Time to Complete Job (hr) t 48 t = w w Number of Workers QY Identif the asmptotes of = _ Variation and Graphs

4 Lesson 2-6 Step 3 Now trace to a negative value of. Continue tracing to the left. Describe what happens to as decreases. Step 4 Does the graph of = _ 10 ever intersect the -ais? If it does, give 2 the coordinates of the point(s) of intersection. If it does not, eplain wh not. Step 5 Does the graph of = _ 10 ever intersect the -ais? If it does, give 2 the coordinates of the point(s) of intersection. If it does not, eplain wh not. Step 6 Trace the graph close to = 0 from both the positive direction and the negative direction. How does the graph behave near = 0? Wh does it behave this wa? The Graph of = k_ 2 Eample 2 eamines the graph of = _ k for two values of k, one 2 positive value and one negative value. Eample 2 a. Graph = 24 _ 2 in a window with { 5 5 } and { }. Sketch the graph on our paper. Then clear the screen and graph = 24 _ 2. b. Describe the smmetr of each graph. Solution a. The graphs below were generated using a graphing utilit. b. If either graph were refl ected over the -ais, the preimage and the image would coincide. So, each graph is smmetric about the -ais. Notice that the inverse-square curve, like a hperbola, has two distinct branches. These two branches, however, do not form a hperbola because the shape and relative position of the branches differ from a hperbola. The domain of ever inverse square function is { 0 }. The range depends on the value of k. When k is positive, the range is { > 0 }. When k is negative, the range is { < 0 }. Graphs of the two tpes of inverse-square curves are shown on the net page. The Graphs of = k_ and = k_ 2 109

5 Chapter 2 II I II I III = k_ 2 for k > 0 Domain { 0 } Range { 0 } IV III = k_ 2 for k < 0 Domain { 0} Range { < 0} Notice that the graph of = _ k has the same vertical and horizontal 2 asmptotes as the graph of = _ k. In general, asmptotes ma be vertical, horizontal, or oblique, but not all graphs have asmptotes. For eample, the parabola = k 2 does not have an asmptotes. When a graph has an asmptote, that asmptote is not part of the graph. Questions COVERING THE IDEAS 1. Wh is 0 not in the domain of the functions with equations = _ k and = _ k? 2 2. For what values of k is the range of the function = _ k the set 2 { < 0 }? 3. Suppose k 0. What shape is the graph of = _ k? In 4 and 5, consider the function f with equation f() = State the domain and range of f. 5. a. Evaluate f( 10) and f(15). b. Graph f in the window and and sketch the graph. 6. Refer to the Condo Care Compan situation and the graph of t = 48 _ w in this lesson. Wh is the domain discrete? 7. Suppose k 0. Is the graph of the equation smmetric to the -ais? a. = k _ b. = k _ 2 8. In which quadrants are the branches of = k _ 2 a. if k is positive? b. if k is negative? IV 110 Variation and Graphs

6 Lesson What happens to the -coordinates of the graph of = 21_ when is positive and is getting closer and closer to 0? 10. Identif the asmptotes of the graph of = 4_. APPLYING THE MATHEMATICS 11. Match each equation with the proper graph. a. = _ 5 2 b. = _ c. = _ i. ii. iii. d. = 5 _ iv. 12. Compare and contrast the graphs of = _ k and = _ k 2 2 when k = 1_ The Doorbell Rang, b Pat Hutchins (1986), tells the stor of 12 cookies to be divided among an increasing number of people. The stor begins with 2 kids who want to share the cookies. Then another kid comes to the door and the must divide the cookies 3 was, then 4, and so on. a. The table at the right shows some values of the function described in the stor. Fill in the missing values. b. Write an equation for a function that contains the coordinate pairs in the table. Eplain the meaning of our variables. c. Eplain wh this function has a discrete domain. 14. Quentin is on a seesaw. He weighs 120 pounds and is sitting 5 feet from the pivot. Recall that the Law of the Lever is d = _ k w, where d is distance from the pivot, w is weight, and k is a constant of variation. a. Find k and write a variation equation for this situation. b. Make a table of 10 different weights and their distances from the pivot that would balance Quentin. c. Plot our values from Part b. d. Is the domain for the contet discrete? Eplain wh or wh not. Number of Kids Number of Cookies Each Kid Gets ? _ 12 5 = 2.4 6? 7 _ ? 9 _ ? The Graphs of = k_ and = k_ 2 111

7 Chapter Consider the functions f and g described b the equations f() = _ 36 and g() = _ a. Use a graphing utilit to graph both functions on the same aes, with the window and Describe some similarities and differences between the graphs. b. Find and compare the average rates of change of each function between = 1 and = 6. c. Find and compare the average rates of change of each function between = 6 and = According to Newton s Law of Gravitation, the force of gravit between two objects obes an inverse-square law with respect to the distance between the two objects. For eample, the force of gravit between Earth and the Voager 1 space probe launched in is approimatel F =, where d is the distance d 2 between Earth and Voager 1 in kilometers, and F is the force in newtons. a. In the summer of 2002, Voager 1 was approimatel km from Earth. What was the force of gravit between Voager 1 and Earth at that time? b. Will Voager 1 ever completel escape Earth s gravit? Eplain our answer. The space probe Voager 1 REVIEW 17. In the graph at the right, parabolas P 1 and P 2 are reflection images of each other over the -ais. If parabola P 1 has equation = 3 2, what is an equation for parabola P 2? (Lesson 2-5) 18. Does the line m graphed below at the right illustrate an eample of direct variation, inverse variation, or neither? Justif our answer. (Lesson 2-4) P 1 P Dominique, a scuba diver 99 feet below the water s surface, inflates a balloon until it has a diameter of 6 inches. The air in the balloon epands as Dominique ascends. When Dominique reaches the surface, the balloon has a diameter of 9.5 inches. (Lesson 2-3) a. Write the ratio of the radius of the balloon at the surface to the radius 99 feet below the surface. b. The volume of the balloon at the surface is how man times the volume of the balloon at 99 feet? c. Eplain how our answer to Part a can be used to answer Part b. m 112 Variation and Graphs

8 Lesson 2-6 Multiple Choice In 20 and 21, which could be a formula for the nth term of the sequence? (Lesson 1-8) 20. 2, 4, 8, 16, 32, A t n = 2n B t n = n 2 C t n = 2 n 21. 2, 9, 28, 65, 126, A t n = 7n - 5 B t n = 7n 2-2 C t n = n Solve for : = 1_ π. (Lesson 1-7) 23. Bob decided to bake a frozen pizza for dinner. It took 10 minutes to preheat the oven to 425 from room temperature of 75. The pizza baked for 17 minutes, and then it took another 30 minutes for the oven to return to room temperature after Bob turned it off. (Lessons 1-4, 1-2) a. Identif the independent and dependent variables in this situation. b. Sketch a graph of the situation. Label the aes and eplain the meaning of our variables. c. Identif the domain and range of the function mapping time onto the temperature of the oven. 24. Multiple Choice The relationship between ABC and PRQ, as shown below, is best described b which of the following? (Previous Course) A Q B C R A congruent B similar C both congruent and similar D neither congruent nor similar P EXPLORATION 25. Use a graphing utilit to graph = _ 10, = _ 10 3, and = _ What pattern do ou notice? What generalization can ou make about the value of n and the graph of = _ 10 n? Is the same generalization true for ever function with equation = _ k n, k 0? QY ANSWERS -ais and -ais The Graphs of = k_ and = k_ 2 113