Determine parabola characteristics:

Step 1: Analyze vertex and focus:

• Vertex: $(-1, 4)$
• Focus: $(-1, 6)$
• Same $x$-coordinate → vertical parabola
• Focus is above vertex → opens upward

Step 2: Find $p$: $p = \text{distance from vertex to focus}$ $p = 6 - 4 = 2$

Step 3: Use standard form for vertical parabola: $(x - h)^2 = 4p(y - k)$

With $h = -1$, $k = 4$, and $p = 2$: $(x-(-1){)}^2=4(2)(y-$

Step 4: Verify key features:

• Vertex: $(-1, 4)$ ✓
• Focus: $(-1, 4 + 2) = (-1, 6)$ ✓
• Directrix: $$

Answer: $(x + 1)^2 = 8(y - 4)$

Alternate form: Solve for $y$: $y = \frac{(x + 1)^2}{8} + 4$

Set up coordinate system:

Place vertex at origin $(0, 0)$, parabola opens upward.

Step 1: Identify a point on the parabola:

• Width at opening: $20$ feet → extends $10$ feet on each side
• Depth: $5$ feet
• Points on rim: $(10, 5)$ and $(-10, 5)$

Step 2: Use standard form with vertex at origin: $x^2 = 4py$

Step 3: Substitute point $(10, 5)$: $10^2 = 4p(5)$ $100 = 20p$

Step 4: Write the equation: $x^2 = 4(5)y$ $x^2 = 20y$

Or: $y = \frac{x^2}{20}$

Step 5: Find focus location: Focus is at $(0, p) = (0, 5)$

Answers:

• Equation: $x^2 = 20y$ or $y = \frac{x^2}{20}$

Practical meaning: All signals hitting the dish will reflect to the focus point $5$ feet above the bottom!

Step 1: Complete the square for x terms: x² - 6x Take half of -6: -3 Square it: (-3)² = 9 x² - 6x = (x - 3)² - 9

Step 2: Complete the square for y terms: y² + 4y Take half of 4: 2 Square it: 2² = 4 y² + 4y = (y + 2)² - 4

Step 3: Substitute back into equation: (x - 3)² - 9 + (y + 2)² - 4 - 3 = 0 (x - 3)² + (y + 2)² - 16 = 0 (x - 3)² + (y + 2)² = 16

Step 4: Identify center and radius: Standard form: (x - h)² + (y - k)² = r² Center: (h, k) = (3, -2) r² = 16, so r = 4

Answer: Center (3, -2), radius 4

Step 1: Identify the vertex from vertex form: y = a(x - h)² + k Vertex: (h, k) = (2, 3)

Step 2: Find p using a = 1/(4p): a = 1/8 1/(4p) = 1/8 4p = 8 p = 2

Step 3: Find the focus: Parabola opens upward (a > 0) Focus is p units above vertex Focus: (2, 3 + 2) = (2, 5)

Step 4: Find the directrix: Directrix is p units below vertex Directrix: y = 3 - 2 = 1

Step 5: Verify: Distance from any point on parabola to focus equals distance to directrix ✓

Answer: Vertex (2, 3), Focus (2, 5), Directrix y = 1