Question

Find the focus and directrix of the parabola y = 1∕2(x +1)^2 + 4.Question 2 options:A) Focus: (–1,41∕2); Directrix: y = 31∕2B) Focus: (1,31∕2); Directrix: y = 41∕2C) Focus: (1,41∕2); Directrix: y = 31∕2D) Focus: (–1,31∕2); Directrix: y = 41∕2

Answer 1

Given the equation:

`y=(1)/(2)\mleft(x+1\mright)^2+4`

• You can identify that it has this form:

`y=a\mleft(x-h\mright)^2+k`

Where its Vertex is:

`(h,k)`

And the Focus is:

`(h,k+(1)/(4a))`

In this case, you can identify that:

`\begin{gathered} h=-1 \\ k=4 \\ \\ a=(1)/(2) \end{gathered}`

Therefore, you can determine that the Focus is:

`(-1,4+(1)/(4\cdot(1)/(2)))=(-1,(9)/(2))`

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:

`4`

- The new numerator will be the Remainder:

`1`

- The denominator does not change.

Then:

`(9)/(2)=4(1)/(2)`

• 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:

`y=k-a`
`y=4-(1)/(2)`
`y=(7)/(2)`

Apply the same procedure shown before, in order to convert the Improper Fraction to a Mixed Number. Hence, you get:

`y=3(1)/(2)`

Therefore, the answer is: Option A.