Full solution

Q. Write the equation in vertex form for the parabola with focus

\left(7, \frac{15}{4}\right)

and directrix

y=-\frac{15}{4}

Identify Parabola Orientation: Identify whether the parabola is vertical or horizontal. $\newline$ Since the directrix is a horizontal line $y = \text{constant}$ , the parabola is vertical.
Vertex Form of Vertical Parabola: Identify the vertex form of a vertical parabola. $\newline$ The vertex form of a vertical parabola is $y = a(x-h)^2 + k$ , where $(h,k)$ is the vertex.
Determine Vertex: Determine the vertex of the parabola. $\newline$ The vertex lies midway between the focus and the directrix. Since the focus is at $(7,\frac{15}{4})$ and the directrix is $y=-(\frac{15}{4})$ , the $y$ -coordinate of the vertex will be the average of $\frac{15}{4}$ and $-\frac{15}{4}$ , which is $0$ . The $x$ -coordinate of the vertex is the same as the $x$ -coordinate of the focus, which is $7$ . Therefore, the vertex is at $(7,0)$ .
Determine Parabola Direction: Determine if the parabola opens upward or downward. $\newline$ The focus is above the directrix, so the parabola opens upward.
Calculate Value of 'a': Calculate the value of 'a'. $\newline$ The distance between the vertex and the focus (or directrix) is the absolute value of the difference in their y-coordinates. This distance is also equal to $\frac{1}{4a}$ . The distance between the focus and the directrix is $\frac{15}{4} - \left(-\frac{15}{4}\right) = \frac{15}{2}$ . Therefore, $\frac{1}{4a} = \frac{15}{2}$ , and $a = \frac{1}{4 \times \frac{15}{2}} = \frac{1}{30}$ .
Write Parabola Equation: Write the equation of the parabola. $\newline$ Substitute $a = \frac{1}{30}$ , $h = 7$ , and $k = 0$ into the vertex form equation. $\newline$ $y = \left(\frac{1}{30}\right)(x-7)^2 + 0$ $\newline$ $y = \left(\frac{1}{30}\right)(x-7)^2$