Given that the focus of the quadratic graph is at (4, 3) and the directrix at y = 13.
This means that the directrix is a horizontal line, meaning that the parabola describing the quadratic graph
must be a regular parabola, where the x
part is squared.
Recall that the equation of a parabola is given by
[tex](x-h)^2=4p(y-k)[/tex]
where (h, k) is the vertex of the parabola and p is the distance between the vertex and the focus or directrix. The value of p is positive if the parabola is facing up and negative if the parabola is facing down.
The distance between the focus and the
directrix is given by 13 - 3 = 10 and the distance from the vertex is half the distance from the directrix, therefore, the distance between the focus and the vertex is 5. Since the focus is below the directrix (i.e. the y value of the focus is
3 while the y value of the directrix is 13), the parabola describing
the quadratic graph is facing down and the value of p is negative.
Hence, p = -5
The vertex of the parabola is the midpoint of the vertical line joining the focus and the directrix.
Thus the vertex of the parabola is the midpoint of the line joining points (4, 3) and (4, 13) and is given by
[tex](h,k)=\left( \frac{4+4}{2} , \frac{3+13}{2} )=(4,8)[/tex]
Thus, the equation of the required graph is given by
[tex](x-4)^2=4(-5)(y-8)=-20(y-8) \\ \\ 20y=-(x-4)^2+160 \\ \\ y=- \frac{1}{20} (x-4)^2+8 \\ \\ f(x)=- \frac{1}{20} (x-4)^2+8[/tex]
Therefore,
the equation of the quadratic graph with a focus of (4, 3) and a directix of y = 13 is
[tex]f(x)=-\frac{1}{20} (x-4)^2+8[/tex]
