Question

equation
$(x - 1)^2 = \frac{1}{4}(y + 1)$
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$(x - 1)^2 = \frac{1}{4}(y + 1)$
thats not it.
find the focus of the parabola.
$(x,y)$
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$(1,1)$
$(1,1)$
correct. good job!
find an equation for the directrix of the parabola.
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$y = - 2$
thats not it.
thats not it. try again. score: 33%

Explanation:

Step1: Recall the standard form of a parabola

The standard form of a parabola that opens up or down is \((x - h)^{2}=4p(y - k)\), where \((h,k)\) is the vertex and \(p\) is the distance from the vertex to the focus (and also from the vertex to the directrix).

Step2: Rewrite the given equation in standard form

Given \((x - 1)^{2}=\frac{1}{4}(y + 1)\), we can rewrite it as \((x - 1)^{2}=4\times\frac{1}{16}(y+ 1)\). So, \(h = 1,k=-1,p=\frac{1}{16}\).

Step3: Find the focus

For a parabola \((x - h)^{2}=4p(y - k)\) that opens up (since \(p=\frac{1}{16}>0\)), the focus is at \((h,k + p)\). Substituting \(h = 1,k=-1,p=\frac{1}{16}\), we get \((1,-1+\frac{1}{16})=(1,-\frac{15}{16})\).

Step4: Find the directrix

The equation of the directrix for a parabola \((x - h)^{2}=4p(y - k)\) is \(y=k - p\). Substituting \(k=-1,p=\frac{1}{16}\), we get \(y=-1-\frac{1}{16}=-\frac{17}{16}\).

Answer:

Focus: \((1,-\frac{15}{16})\)
Directrix: \(y =-\frac{17}{16}\)