1016 CHAPTER 10 Analytic Geometry EXAMPLE 1 Using Asymptotes to Graph a Hyperbola Graph x2 25 - y2 49 = 1. Sketch the asymptotes, and find the coordinates of the vertices and foci. Give the domain and range. ALGEBRAIC SOLUTION For this hyperbola, a = 5 and b = 7. With these values, y = { b a x becomes y = { 7 5 x. Asymptotes If we choose x = 5, then y = {7. Choosing x = -5 also gives y = {7. These four ordered pairs—15, 72, 15, -72, 1-5, 72, and 1-5, -72—are the coordinates of the corners of the rectangle shown in Figure 30. GRAPHING CALCULATOR SOLUTION The graph of a hyperbola is not the graph of a function. We solve for y in x 2 25 - y2 49 = 1 to obtain equations of the two functions y = { 7 5 2x2 - 25. (–5, 7) V(5, 0) (5, 7) (–5, –7) (5, –7) – = 1 25 x2 49 y2 F9(–Ë74 , 0) F(Ë74 , 0) x y V9(–5, 0) Figure 30 The extended diagonals of this rectangle, called the fundamental rectangle, are the asymptotes of the hyperbola. Because a = 5, the vertices of the hyperbola are 15, 02 and 1-5, 02, as shown in Figure 30. We find the foci by letting c2 = a2 + b2 = 25 + 49 = 74, so c = 274. Therefore, the foci are A 274, 0B and A -274, 0B. The domain is 1-∞, -54 ´35, ∞2, and the range is 1-∞, ∞2. y4 = − x −10 −10 10 10 7 5 y1 = Ëx 2 − 25 7 5 y3 = x 7 5 y2 = − Ëx 2 − 25 7 5 The graph of y1 is the upper portion of each branch of the hyperbola shown in Figure 31, and the graph of y2 is the lower portion of each branch. Alternatively, we could enter y2 = -y1 to obtain the part of the graph below the x-axis. The asymptotes are also shown. We can use tracing to observe how the branches of the hyperbola approach the asymptotes. S Now Try Exercise 9. Figure 31 While a 7b for an ellipse, examples show that for hyperbolas, it is possible that a 7b, a 6b, or a = b. If the foci of a hyperbola are on the y-axis, the equation of the hyperbola has the form y2 a2 − x2 b2 =1, with asymptotes y = t a b x. In either case, whether transverse axis on the x-axis or transverse axis on the y-axis, a2 is chosen as the denominator of the leading term in the equation of a hyperbola written in standard form. NOTE When graphing hyperbolas, remember that the fundamental rectangle and the asymptotes are not actually parts of the graph. They are simply aids in sketching the graph.
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