### About Conic Sections: The Parabola:

A parabola is the set of all points that are equidistant from a fixed point called the focus and a fixed line called the directrix.

Test Objectives
• Demonstrate the ability to find the focus of a parabola
• Demonstrate the ability to find the directrix of a parabola
• Demonstrate the ability to write a parabola in vertex form
Conic Sections: The Parabola Practice Test:

#1:

Instructions: Identify the focus and directrix.

$$a)\hspace{.2em}y=\frac{1}{2}x^2 + 8x + 26$$

$$b)\hspace{.2em}x=\frac{1}{4}y^2 - \frac{7}{2}y + \frac{69}{4}$$

#2:

Instructions: Write in vertex form.

$$a)\hspace{.2em}Vertex:(5, -4)$$ $$Focus: \left(5, -\frac{31}{8}\right)$$

$$b)\hspace{.2em}Vertex:(-6, -6)$$ $$Focus: \left(-6, -\frac{23}{4}\right)$$

#3:

Instructions: Write in vertex form.

$$a)\hspace{.2em}Vertex:(3, 4)$$ $$Focus: \left(3, \frac{17}{4}\right)$$

$$b)\hspace{.2em}Vertex:(-4, 6)$$ $$Focus: \left(-\frac{127}{32}, 6\right)$$

#4:

Instructions: Write in vertex form.

$$a)\hspace{.2em}Vertex:(-10, -1)$$ $$Focus: \left(-\frac{15}{2}, -1\right)$$

$$b)\hspace{.2em}Vertex:(-3, 6)$$ $$Focus: \left(-\frac{25}{8}, 6\right)$$

#5:

Instructions: Write in vertex form.

$$a)\hspace{.2em}Vertex:(-5, 10)$$ $$Point: \left(0, 9\right)$$ $$\text{Horizontal}$$

$$b)\hspace{.2em}Vertex:(-10, -4)$$ $$Point: \left(-8, -20\right)$$ $$\text{Vertical}$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}\text{focus}: \left(-8, -\frac{11}{2}\right)$$ $$\text{directrix}: y=-\frac{13}{2}$$

$$b)\hspace{.2em}\text{focus}: (6,7)$$ $$\text{directrix}: x=4$$

#2:

Solutions:

$$a)\hspace{.2em}y=2(x - 5)^2 - 4$$

$$b)\hspace{.2em}y=(x + 6)^2 - 6$$

#3:

Solutions:

$$a)\hspace{.2em}y=(x - 3)^2 + 4$$

$$b)\hspace{.2em}x=8(y - 6)^2 - 4$$

#4:

Solutions:

$$a)\hspace{.2em}x=\frac{1}{10}(y + 1)^2 - 10$$

$$b)\hspace{.2em}x=-2(y - 6)^2 - 3$$

#5:

Solutions:

$$a)\hspace{.2em}x=5(y - 10)^2 - 5$$

$$b)\hspace{.2em}y=-4(x + 10)^2 - 4$$