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A parabola has its focus at (–2, 7) and its vertex at (–2, –2). What is the equation of the parabolas directrix?

parabola: \(y=ax^2+bx+c\)

focus: \( (x_f = -2,\ y_f = 7)\)

vertex:  \((x_v = -2,\ y_v = -2)\)

directrix: \(y = c - \frac{b^2+1}{4a}\)

\(\boxed{~ \begin{array}{rclrcl} x_f &=& -\frac{b}{2a} & y_f &=& c - \frac{b^2-1}{4a} \\ x_v &=& -\frac{b}{2a} & y_v &=& c - \frac{b^2}{4a} \\ \end{array} ~}\)

1. a?

\(\begin{array}{rcll} y_f = c - \frac{b^2-1}{4a} &=& c - \frac{b^2}{4a} +\frac{1}{4a} \\ y_v &=& c - \frac{b^2}{4a} \\\\ y_f-y_v &=& \frac{1}{4a} \\ a &=& \frac{1}{4\cdot (y_f-y_v)} \\ a &=& \frac{1}{4\cdot (7-(-2))} \\ a &=& \frac{1}{4\cdot 9} \\ \mathbf{a} &\mathbf{=}& \mathbf{\frac{1}{36}} \\ \end{array}\)

2. b?

\(\begin{array}{rcl} x_v &=& -\frac{b}{2a} \\ \frac{b}{2a} &=& -x_v \\ b &=& -2\cdot a\cdot x_v \\ b &=& -2\cdot \frac{1}{36} \cdot (-2) \\ b &=& 4\cdot \frac{1}{36} \\ \mathbf{b} &\mathbf{=}& \mathbf{\frac{1}{9}} \\ \end{array}\)

3. c?

\(\begin{array}{rcl} y_v &=& c - \frac{b^2}{4a} \\ \frac{b^2}{4a} + y_v &=& c \\ c &=& \frac{b^2}{4a} + y_v \\ c &=& \frac{(\frac{1}{9})^2}{4\cdot (\frac{1}{36})} + (-2) \\ c &=& \frac{1}{9}-2 \\ \mathbf{c} &\mathbf{=}& \mathbf{-\frac{17}{9}} \\ \end{array}\)

The equation of the parabola is \(y=\frac{1}{36}x^2+\frac{1}{9}x-\frac{17}{9}\)

4. directrix

\(\begin{array}{rcl} y &=& c - \frac{b^2+1}{4a} \\ y &=& -\frac{17}{9} - \frac{(\frac{1}{9})^2+1}{4\cdot \frac{1}{36} } \\ y &=& -\frac{17}{9} - 9\cdot (\frac{1}{81}+1) \\ y &=& -\frac{17}{9} -\frac19 - 9 \\ y &=& -2 - 9 \\ \mathbf{y} &\mathbf{=}& \mathbf{-11} \\ \end{array} \)

The equation of the parabolas directrix is y = -11

A parabola has its focus at (–2, 7) and its vertex at (–2, –2). What is the equation of the parabolas directrix?

Draw a skech and you can almost see the answer immediately.

The focus and vertex lie n the vertical line x=-2

The vertex is below the focus so it is concave up.

The focus and the directrix are 7--2=9 units apart.

so the directrix will be 9 units below the vertex.

-2-9=-11

The directrix is     y= -11