Given the equation:
[tex]y=\frac{1}{2}\mleft(x+1\mright)^2+4[/tex]• You can identify that it has this form:
[tex]y=a\mleft(x-h\mright)^2+k[/tex]Where its Vertex is:
[tex](h,k)[/tex]And the Focus is:
[tex](h,k+\frac{1}{4a})[/tex]In this case, you can identify that:
[tex]\begin{gathered} h=-1 \\ k=4 \\ \\ a=\frac{1}{2} \end{gathered}[/tex]Therefore, you can determine that the Focus is:
[tex](-1,4+\frac{1}{4\cdot\frac{1}{2}})=(-1,\frac{9}{2})[/tex]In order to write the y-coordinate of the Focus as a Mixed Numbers, you need to:
- Divide the numerator by the denominator.
- The Quotient will be the whole number part:
[tex]4[/tex]- The new numerator will be the Remainder:
[tex]1[/tex]- The denominator does not change.
Then:
[tex]\frac{9}{2}=4\frac{1}{2}[/tex]• In order to find the Directrix, you need to remember that, by definition, the Directrix has the same distance from the vertex that the Focus of the parabola is. Therefore:
[tex]y=k-a[/tex][tex]y=4-\frac{1}{2}[/tex][tex]y=\frac{7}{2}[/tex]Apply the same procedure shown before, in order to convert the Improper Fraction to a Mixed Number. Hence, you get:
[tex]y=3\frac{1}{2}[/tex]Therefore, the answer is: Option A.
