| Introduction | | | | are given by (h +/- a, k). |
| The following will serve as a brief overview of | | | | On the other hand, an ellipse centered at (h, k) is |
| conic sections or in other words, the functions | | | | given by [(x - h)^2 / (b^2)] + [(y - k)^2 / (a^2)] |
| and graphs associated with the parabola, the | | | | = 1 when the major axis is vertical. Here, the foci |
| circle, the ellipse, and the hyperbola. Initially, it | | | | are given by (h, k +/- c) and the vertices are |
| should be noted that these functions are named | | | | given by (h, k+/- a). |
| conic sections since they represent the various | | | | Note that in both types of standard equations for |
| ways in which a plane can intersect with a pair of | | | | the ellipse, a > b > 0. Also, c^2 = a^2 - b^2. It is |
| cones. | | | | important to note that 2a always represents the |
| The Parabola | | | | length of the major axis and 2b always |
| The first conic section usually studied is the | | | | represents the length of the minor axis. |
| parabola. The equation of a parabola with a | | | | The Hyperbola |
| vertex at (h, k) and a vertical axis of symmetry | | | | The hyperbola is probably the most difficult conic |
| is defined as (x - h)^2 = 4p(y - k). Note that if p | | | | section to draw and understand. By memorizing |
| is positive, the parabola opens upward and if p is | | | | the following equations and practicing by sketching |
| negative, it opens downward. For this type of | | | | graphs, one can master even the most difficult |
| parabola, the focus is centered at the point (h, k | | | | hyperbola problem. |
| + p) and the directrix is a line found at y = k - p. | | | | To start, the standard equation of a hyperbola |
| On the other hand, the equation of a parabola | | | | with center (h, k) and a horizontal transverse axis |
| with a vertex at (h, k) and a horizontal axis of | | | | is given by [(x - h)^2/a^2] - [(y - k)^2/b^2] = 1. |
| symmetry is defined as (y - k)^2 = 4p(x - h). | | | | Note that the terms of this equation are |
| Note that if p is positive, the parabola opens to | | | | separated by a minus sign instead of a plus sign |
| the right and if p is negative, it opens to the left. | | | | with the ellipse. Here, the foci are given by the |
| For this type of parabola, the focus is centered at | | | | points (h +/- c, k), thevertices are given by the |
| the point (h + p, k) and the directrix is a line | | | | points (h +/- a, k) and the asymptotes are |
| found at x = h - p. | | | | represented by y = +/- (b/a)(x - h) +k. |
| The Circle | | | | Next, the standard equation of a hyperbola with |
| The next conic section to be analyzed is the circle. | | | | center (h, k) and a vertical transverse axis is |
| The equation of a circle of radius r centered at | | | | given by [(y- k)^2/a^2] - [(x - h)^2/b^2] = 1. |
| the point (h, k) is given by (x - h)^2 + (y - k)^2 | | | | Note that the terms of this equation are |
| = r^2. | | | | separated by a minus sign instead of a plus sign |
| The Ellipse | | | | with the ellipse. Here, the foci are given by the |
| The standard equation of an ellipse centered at (h, | | | | points (h, k +/- c), the vertices are given by the |
| k) is given by [(x - h)^2/a^2] + [(y - k)^2/b^2] | | | | points (h, k +/- a) and the asymptotes are |
| = 1 when the major axis is horizontal. In this case, | | | | represented by y = +/- (a/b)(x - h) + k. |
| the foci are given by (h +/- c, k) and the vertices | | | | |