Sketch the graph of each parabola and label all its parts. (5 points each)
5.x²=-19y
6.(x+5)²=6(y- 3)

### Problem 5: \( x^2 = -19y \)

This equation is a **vertical parabola** in the form \( x^2 = 4py \), where:
- **Vertex:** The vertex is at the origin \( (0, 0) \), since the equation does not contain any horizontal or vertical shifts.
- **Direction:** Since \( 4p = -19 \), the parabola opens **downward** (because of the negative sign).
- **Focus:** From \( 4p = -19 \), we can solve for \( p \):  
  \( p = -\frac{19}{4} \approx -4.75 \).  
  The focus is at \( (0, p) = (0, -4.75) \).
- **Directrix:** The directrix is the line \( y = -p = \frac{19}{4} \approx 4.75 \).
- **Axis of Symmetry:** The axis of symmetry is the vertical line \( x = 0 \) (the y-axis).

### Sketch:
- The vertex is at the origin \( (0, 0) \).
- The parabola opens downward.
- The focus is at \( (0, -4.75) \), and the directrix is the horizontal line \( y = 4.75 \).

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### Problem 6: \( (x+5)^2 = 6(y - 3) \)

This is also a **vertical parabola**, but with horizontal and vertical shifts. It is in the form \( (x - h)^2 = 4p(y - k) \), where:
- **Vertex:** The vertex is at \( (-5, 3) \) (from \( x+5 \) and \( y - 3 \)).
- **Direction:** Since \( 4p = 6 \), the parabola opens **upward** (because the coefficient is positive).
- **Focus:** From \( 4p = 6 \), solve for \( p \):  
  \( p = \frac{6}{4} = \frac{3}{2} = 1.5 \).  
  The focus is \( 1.5 \) units above the vertex, so the focus is at \( (-5, 3 + 1.5) = (-5, 4.5) \).
- **Directrix:** The directrix is \( y = 3 - 1.5 = 1.5 \).
- **Axis of Symmetry:** The axis of symmetry is the vertical line \( x = -5 \).

### Sketch:
- The vertex is at \( (-5, 3) \).
- The parabola opens upward.
- The focus is at \( (-5, 4.5) \), and the directrix is the horizontal line \( y = 1.5 \).

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Would you like more details on how to graph these, or any questions?

### 5 Related Questions:
1. How do you determine the direction a parabola opens from its equation?
2. What is the relationship between the focus and directrix in a parabola?
3. How does shifting the vertex affect the equation of a parabola?
4. How do you find the focus and directrix from the standard form of a parabola equation?
5. How does changing the coefficient of \( y \) or \( x \) affect the width of the parabola?

#### Tip:
For parabolas in the form \( x^2 = 4py \), the value of \( p \) controls how "wide" or "narrow" the parabola is—larger \( |p| \) values make the parabola wider, while smaller \( |p| \) values make it narrower.

Sketch the graph of each parabola and label all its parts. (5 points each)
5. x² = -19y
6. (x+5)² = 6(y - 3)

Learn how to graph parabolas with detailed solutions for x² = -19y and (x+5)² = 6(y - 3). This guide covers key concepts such as the vertex, focus, directrix, and axis of symmetry. Suitable for students in Grades 9-12.

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