Question

A list of attributes for a parabola are shown. The directrix of the parabola is represented by the line y=1. The line y=7 passes through the focus of the parabola. The axis of symmetry is represented by the line x=-3.5.

Answer 1

The high school math question involves finding the standard equation of a parabola given its directrix, focus, and axis of symmetry. The vertex is calculated to be (-3.5, 4), and with the focus lying above the directrix, the equation is derived as (x + 3.5)^2 = 12(y - 4).

Step-by-step explanation:

The question given pertains to parabolas, which are geometric shapes defined by their algebraic properties in a coordinate plane. The information regarding the directrix (y=1), focus (y=7), and axis of symmetry (x=-3.5) allows us to determine the standard equation of the parabola.

To find the parabola's equation, we use the focus-directrix property of parabolas. The distance from any point on the parabola to the focus equals the distance from that point to the directrix. Given that the focus lies on the line y=7 and the directrix is y=1, we know the vertex is the midpoint between them, hence at y=4. With the axis of symmetry at x=-3.5, we can now deduce the vertex of the parabola is at (-3.5, 4). The parabola opens upwards since the focus is above the directrix.

Now, with the information provided, we can write the formula for the parabola as (x + 3.5)^2 = 4p(y - 4), where p is the distance between the focus and the vertex. Since this distance is 3, we have (x + 3.5)^2 = 4*3(y - 4) or (x + 3.5)^2 = 12(y - 4).