Algebra II worksheet focusing on parabolas, circles, and ellipses with various problems and a graphing grid.

Explanation:
Let’s go through each problem one by one, carefully and step by step.

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1. Find the focus of the parabola: $ y^2 = -8x $

This is a horizontal parabola (opens left or right) because $ y $ is squared.

Standard form for a horizontal parabola with vertex at origin:
$$ y^2 = 4px $$
- If $ p > 0 $, opens right; if $ p < 0 $, opens left.
- Focus is at $ (p, 0) $
- Directrix is $ x = -p $

Given: $ y^2 = -8x $
Compare to $ y^2 = 4px $:
So $ 4p = -8 \Rightarrow p = -2 $

→ Focus is at $ (p, 0) = (-2, 0) $

✔ Answer: $ (-2, 0) $

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2. Identify the focus and directrix of the parabola given by $ y^2 = -4x $

Same idea: compare to $ y^2 = 4px $

$ 4p = -4 \Rightarrow p = -1 $

→ Focus: $ (p, 0) = (-1, 0) $
→ Directrix: $ x = -p = 1 $

✔ Focus: $ (-1, 0) $, Directrix: $ x = 1 $

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3. Identify the focus and directrix of the parabola given by $ x^2 = 12y $

Now $ x $ is squared → vertical parabola (opens up/down).
Standard form: $ x^2 = 4py $

Compare: $ 4p = 12 \Rightarrow p = 3 $

Since $ p > 0 $, opens upward.

→ Focus: $ (0, p) = (0, 3) $
→ Directrix: $ y = -p = -3 $

✔ Focus: $ (0, 3) $, Directrix: $ y = -3 $

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4. Graph the parabola $ y^2 - 8x = 0 $. Include the vertex, focus, directrix, and four points other than the vertex.

First rewrite: $ y^2 = 8x $

This is horizontal, opens right.

Standard form: $ y^2 = 4px $ → $ 4p = 8 \Rightarrow p = 2 $

→ Vertex: $ (0, 0) $
→ Focus: $ (p, 0) = (2, 0) $
→ Directrix: $ x = -p = -2 $

Now find 4 points (other than vertex):

Pick values for $ y $, solve for $ x $:

- $ y = 2 $: $ x = \frac{y^2}{8} = \frac{4}{8} = 0.5 $ → point $ (0.5, 2) $
- $ y = -2 $: $ x = 0.5 $ → $ (0.5, -2) $
- $ y = 4 $: $ x = \frac{16}{8} = 2 $ → $ (2, 4) $
- $ y = -4 $: $ x = 2 $ → $ (2, -4) $

You can plot these on the grid.

✔ Vertex: $ (0,0) $, Focus: $ (2,0) $, Directrix: $ x = -2 $, Points: $ (0.5, 2), (0.5, -2), (2, 4), (2, -4) $

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5. Write the standard form of the equation of the parabola with its vertex at $ (0,0) $ and focus at $ (0, -4) $

Vertex at origin, focus at $ (0, -4) $ → vertical parabola, opens downward.

Distance from vertex to focus = $ |p| = 4 $, and since it opens down, $ p = -4 $

Standard form: $ x^2 = 4py $

Plug in $ p = -4 $:
$ x^2 = 4(-4)y = -16y $

✔ Equation: $ x^2 = -16y $

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6. Write the standard form of the equation of the parabola with its vertex at $ (0,0) $ and directrix $ y = 5 $

Directrix is horizontal line $ y = 5 $, vertex at $ (0,0) $, so parabola opens down (since directrix is above vertex).

For vertical parabola: directrix is $ y = -p $, so:

$ -p = 5 \Rightarrow p = -5 $

Then standard form: $ x^2 = 4py = 4(-5)y = -20y $

✔ Equation: $ x^2 = -20y $

Wait — double-check:
If directrix is $ y = 5 $, and vertex is $ (0,0) $, then distance from vertex to directrix is 5 units upward → focus must be 5 units downward → focus at $ (0, -5) $, so $ p = -5 $. Yes.

So $ x^2 = 4(-5)y = -20y $. Correct.

✔ $ x^2 = -20y $

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7. Write the standard form of the equation of the parabola with its vertex at $ (0,0) $ and directrix $ x = 2 $

Directrix is vertical line $ x = 2 $, vertex at $ (0,0) $, so parabola opens left (directrix is to the right of vertex).

For horizontal parabola: directrix is $ x = -p $, so:

$ -p = 2 \Rightarrow p = -2 $

Standard form: $ y^2 = 4px = 4(-2)x = -8x $

✔ Equation: $ y^2 = -8x $

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8. Suppose a parabola has vertex $ (0,0) $ and the distance from the vertex to the focus is 5 units. How many possible parabolas fit this description? Write the equations of all the possible parabolas that fit this description.

Distance from vertex to focus = 5 → $ |p| = 5 $, so $ p = 5 $ or $ p = -5 $

But orientation matters: parabola can open up, down, right, or left.

- Vertical opening: uses form $ x^2 = 4py $
- $ p = 5 $: opens up → $ x^2 = 20y $
- $ p = -5 $: opens down → $ x^2 = -20y $

- Horizontal opening: uses form $ y^2 = 4px $
- $ p = 5 $: opens right → $ y^2 = 20x $
- $ p = -5 $: opens left → $ y^2 = -20x $

So there are 4 possible parabolas.

✔ Equations:
- $ x^2 = 20y $
- $ x^2 = -20y $
- $ y^2 = 20x $
- $ y^2 = -20x $

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Now compile all final answers clearly:

Final Answer:
1. $ (-2, 0) $
2. Focus: $ (-1, 0) $, Directrix: $ x = 1 $
3. Focus: $ (0, 3) $, Directrix: $ y = -3 $
4. Vertex: $ (0,0) $, Focus: $ (2,0) $, Directrix: $ x = -2 $, Points: $ (0.5, 2), (0.5, -2), (2, 4), (2, -4) $
5. $ x^2 = -16y $
6. $ x^2 = -20y $
7. $ y^2 = -8x $
8. 4 possible parabolas: $ x^2 = 20y $, $ x^2 = -20y $, $ y^2 = 20x $, $ y^2 = -20x $