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534 Worked-Out Solutions to Exercises: 11-19

8 Here s the original equation again, for reference: x 2 y 2 + 2x + 2y = 4 As in Problem 7, we must rely on our algebra experience to see how this can be transformed into an equation that s in the standard form for a conic section We can split it into a difference between two trinomials as (x 2 + 2x + 1) (y 2 2y + 1) = 4 In this equation, the parentheses are necessary! The trinomials can be factored exactly as they were in the solution to Problem 7; when we do that, we obtain (x + 1)2 (y 1)2 = 4 We can divide through by 4, getting (x + 1)2 /4 (y 1)2 /4 = 1 This equation is in the standard form for a hyperbola The general version, as we ve learned, is (x x0)2 /a2 (y y0)2 /b2 = 1 where (x0,y0) are the coordinates of the center, a is the length of the horizontal semi-axis, and b is the length of the vertical semi-axis In this case, we have x0 = 1 y0 = 1 a = 41/2 = 2 b = 41/2 = 2 The center is at ( 1,1) The horizontal and vertical semi-axes are both 2 units long With this information, we can sketch the graph as shown in Fig B-11 9 Here s the original equation again, for reference: x 2 3x y + 3 = 1 We can subtract 1 from each side to get x 2 3x y + 2 = 0 Adding y to each side and then transposing the sides left-to-right yields y = x 2 3x + 2

Here are the stages of the SDLC that we will explore more thoroughly in Part Two. The concept stage. Not every bright idea is a good idea. The concept stage is our chance to evaluate every bright idea. When we do, we can discard the ones that will not work early, saving time and money. We create a context diagram, which illustrates the role of the new product or service in our company. We can create a basic plan for the good ideas, and use that plan to explain our ideas, build support for the project, and secure funding. The gate at the end of the concept stage is used to stop bad ideas from becoming projects that waste money. We want and expect many ideas to be cancelled at the end of the concept stage, and only projects with real value to move forward. The analysis stage. This stage is all about the customer and the stakeholders. We seek to de ne what they want at all levels. We document the purpose and value of the system, the goals of the system, and a detailed description of functions and features. We add details to the context diagram. We review the project plans with all stakeholders,

Look back at Fig 3-2 If you ponder this graph for awhile, you might suspect that the indicated equations aren t the only ones that can represent these lines You might ask, If we allow r to range over all the real numbers, both positive and negative, can t the line for q = p /3 also be represented by other equations such as q = 4p /3 or q = 2p /3 Can t the line representing the q = 7p /8 also be represented by q = 15p /8 or q = p /8 The answers to these questions are Yes When we see an equation of the form q = a representing a straight line through the origin in polar coordinates, we can add any integer multiple of p to the constant a, and we get another equation whose graph is the same line In more formal terms, a particular line q = a through the origin can be represented by q = kp a where k is any integer and a is a real-number constant

Here s a challenge!