992 CHAPTER 10 Analytic Geometry Ellipse Axis Circle Parabola Axis Hyperbola Axis Figure 1 Conic Sections Parabolas, circles, ellipses, and hyperbolas form a group of curves known as conic sections, because they are the results of intersecting a cone with a plane. See Figure 1. 10.1 Parabolas ■ Conic Sections ■ Horizontal Parabolas ■ Geometric Definition and Equations of Parabolas ■ An Application of Parabolas Horizontal Parabolas We know that the graph of the equation y = a1x - h22 + k is a parabola with vertex 1h, k2 and the vertical line x = h as its axis of symmetry. If we subtract k from each side and interchange the roles of x - h and y - k, the new equation also has a parabola as its graph. y - k = a1x - h22 Subtract k. (1) x - h = a1y - k22 Interchange the roles of x - h and y - k. (2) While the graph of y - k = a1x - h22 has a vertical axis of symmetry, the graph of x - h = a1y - k22 has a horizontal axis of symmetry. The graph of the first equation is the graph of a function (specifically a quadratic function), while the graph of the second equation is not. Its graph fails the vertical line test. ▼ ▼ Parabola with Horizontal Axis of Symmetry The parabola with vertex 1h, k2 and the horizontal line y = k as axis of symmetry has an equation of the following form. x −h =a1y −k22 The parabola opens to the right if a 70 and to the left if a 60. x y y = x2 x = y2 y = x –2 2 4 2 4 –2 Figure 2 NOTE When the vertex 1h, k2 is 10, 02 and a = 1 in y - k = a1x - h22 (1) and x - h = a1y - k22, (2) the equations y = x2 and x = y2, respectively, result. See Figure 2. The graphs are mirror images of each other with respect to the line y = x.
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