Answer:
Option D - [tex]f(x)=2x^2+34+16x[/tex]
Step-by-step explanation:
Given : Focus [tex]F=(-4,\frac{17}{8})[/tex] and a directex [tex]y=\frac{15}{8}[/tex]
To find : What is the equation of the quadratic graph with a focus and directrix?
Solution :
The general form of the quadratic equation is [tex](x-h)^2=4p(y-k)[/tex]
Where, Focus of the equation is [tex]F=(h,k+p)[/tex] and
Directrix is [tex]y=k-p[/tex]
Comparing the given focus and directrix with the general one.
[tex](-4,\frac{17}{8})=(h,k+p)[/tex]
h=-4 and
[tex]k+p=\frac{17}{8}[/tex] ....(1)
[tex]k-p=\frac{15}{8}[/tex] .....(2)
Add equation (1) and (2)
[tex]2k=4[/tex]
[tex]k=2[/tex]
Substitute in equation (1)
[tex]2+p=\frac{17}{8}[/tex]
[tex]p=\frac{17}{8}-2[/tex]
[tex]p=\frac{17-16}{8}[/tex]
[tex]p=\frac{1}{8}[/tex]
Now, We get h=-4 , k=2 and [tex]p=\frac{1}{8}[/tex]
Substitute in the general form,
[tex](x-(-4))^2=4(\frac{1}{8})(y-2)[/tex]
[tex]x^2+16+8x=\frac{1}{2}(y-2)[/tex]
[tex]2x^2+32+16x=y-2[/tex]
[tex]2x^2+32+16x=y-2[/tex]
i.e, The equation of the quadratic graph is [tex]f(x)=2x^2+34+16x[/tex]
Therefore, Option D is correct.
