Final answer:
The question asks for the vertex, focus, and directrix of the parabola -(1)/(20)(y-4)²=(x-2). The vertex of the parabola is (2, 4), the focus is (-3, 4), and the directrix is the vertical line x = 7.
Step-by-step explanation:
The student is asking for the vertex, focus, and directrix of a parabola given in the form -(1)/(20)(y-4)²=(x-2). This equation is a transformed version of the standard form of a parabola, which is y = ax² for a parabola that opens horizontally. The negative coefficient indicates that the parabola opens to the left, and the fractions suggest the horizontal compression and vertical shift of the graph.
To find the vertex, we set the squared terms to zero, which results in vertex (2, 4). To find the focus, we use the fact that the distance from the vertex to the focus is 1/4a, where a is the coefficient of the squared term. Since a in our equation is -1/20, the distance from the vertex to the focus is -20/4 which is -5. This gives us the focus (2-5, 4) or (-3,4). The directrix, then, is a vertical line that is the same distance from the vertex as the focus but in the opposite direction, giving us the equation x = 2+5 or x = 7 as the directrix.