A parabola is a conic section. It is a slice of a right cone parallel to one side (a generating line) of the cone. Like the circle, the parabola is a quadratic relation, but unlike the circle, either x will be squared or y will be squared, but not both. You worked with parabolas in Algebra 1 when you graphed quadratic equations. We will now be investigating the conic form of the parabola equation to learn more about the parabola's graph.
y = ax2 + bx + c from your study of quadratics. And, of course, these remain popular equation forms of a parabola. But, if we examine a parabola in relation to its focal point (focus) and directrix, we can determine more information about the parabola. We are now going to look more closely at the coefficient of the x2 term to see what additional information it can tell us about the graph of the parabola. Keep in mind that all information you already know about parabolas remains true!
You remember the vertex form of a parabola as being y = a(x - h)2 + k where (h, k) is the vertex of the parabola. If we let the coefficient of x2 (or a) = and perform some algebraic maneuvering, we can get the next equation.
S u m m i n g U p: (p is distance from vertex to focus)
Given x2 = 16y, state whether the parabola opens upward, downward, right or left, and state the coordinates of the focus.
Given the parabola, (x - 3)2 = -8(y - 2), state whether the parabola opens upward, downward, right or left, and state the coordinates of the vertex, the focus, and the equation of the directrix.
Find the focus and directrix of the parabola whose equation is x2 - 6x + 3y + 18 = 0.
Write the equation of a parabola whose focus (-2,1) and whose directrix is x = -6.
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