The equation of the parabola with the given vertex and directrix in vertex form is y = (1/16)( x + 5 )² - 9.
Hence, option C is the correct answer.
What is the equation of the parabola?
Given the data in the question;
- Vertex of the parabola: ( -5, -9 )
- h = -5
- k = -9
- Directrix of the parabola: y = -13
To find the equation, we use the equation of the parabola that opens up or down since the directrix ( y = -13 ) is vertical.
The equation is expressed as;
( x - h )² = 4p( y - k )
First, we find the distance from the focus to the vertex.
|p| is the distance rom the focus to the vertex and from the vertex to the directrix.
p = -9 + 13
p = 4
We substitute the values into the equation;
( x - h )² = 4p( y - k )
( x - (-5) )² = 4(4)( y - (-9) )
( x + 5 )² = 16( y + 9 )
Multiply both side by 1/16
(1/16)( x + 5 )² = y + 9
Make y the subject of the formula
(1/16)( x + 5 )² - 9 = y
y = (1/16)( x + 5 )² - 9
The equation of the parabola with the given vertex and directrix in vertex form is y = (1/16)( x + 5 )² - 9.
Hence, option C is the correct answer.
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