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
#1:
Instructions: Identify the focus and directrix.
$$a)\hspace{.2em}y=\frac{1}{2}x^2 + 8x + 26$$
$$b)\hspace{.2em}x=\frac{1}{4}y^2 - \frac{7}{2}y + \frac{69}{4}$$
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#2:
Instructions: Write in vertex form.
$$a)\hspace{.2em}Vertex:(5, -4)$$ $$Focus: \left(5, -\frac{31}{8}\right)$$
$$b)\hspace{.2em}Vertex:(-6, -6)$$ $$Focus: \left(-6, -\frac{23}{4}\right)$$
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#3:
Instructions: Write in vertex form.
$$a)\hspace{.2em}Vertex:(3, 4)$$ $$Focus: \left(3, \frac{17}{4}\right)$$
$$b)\hspace{.2em}Vertex:(-4, 6)$$ $$Focus: \left(-\frac{127}{32}, 6\right)$$
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#4:
Instructions: Write in vertex form.
$$a)\hspace{.2em}Vertex:(-10, -1)$$ $$Focus: \left(-\frac{15}{2}, -1\right)$$
$$b)\hspace{.2em}Vertex:(-3, 6)$$ $$Focus: \left(-\frac{25}{8}, 6\right)$$
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#5:
Instructions: Write in vertex form.
$$a)\hspace{.2em}Vertex:(-5, 10)$$ $$Point: \left(0, 9\right)$$ $$\text{Horizontal}$$
$$b)\hspace{.2em}Vertex:(-10, -4)$$ $$Point: \left(-8, -20\right)$$ $$\text{Vertical}$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}\text{focus}: \left(-8, -\frac{11}{2}\right)$$ $$\text{directrix}: y=-\frac{13}{2}$$
$$b)\hspace{.2em}\text{focus}: (6,7)$$ $$\text{directrix}: x=4$$
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#2:
Solutions:
$$a)\hspace{.2em}y=2(x - 5)^2 - 4$$
$$b)\hspace{.2em}y=(x + 6)^2 - 6$$
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#3:
Solutions:
$$a)\hspace{.2em}y=(x - 3)^2 + 4$$
$$b)\hspace{.2em}x=8(y - 6)^2 - 4$$
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#4:
Solutions:
$$a)\hspace{.2em}x=\frac{1}{10}(y + 1)^2 - 10$$
$$b)\hspace{.2em}x=-2(y - 6)^2 - 3$$
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#5:
Solutions:
$$a)\hspace{.2em}x=5(y - 10)^2 - 5$$
$$b)\hspace{.2em}y=-4(x + 10)^2 - 4$$