Identify the vertex, focus, axis of symmetry, and directrix of each. Then sketch the graph.

1. y=-(x+3)^2)-1
2. x=-(1/4)((y+2)^2)
3. -(1/3)(x-3)=(y+5)^2
4. -(y+2)=(x-2)^2
5. 3y+4x=-(2x^2)-14
6. x=-2(y+2)^2

Let's analyze each equation to identify the vertex, focus, axis of symmetry, and directrix. Then, we will describe how the graph looks for each equation.

### 1. \( y = -(x + 3)^2 - 1 \)

This is a parabola that opens downward because of the negative sign in front of the squared term.

- **Vertex:** The vertex form of the equation is \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex. Here, the vertex is \((-3, -1)\).
- **Focus:** The distance from the vertex to the focus is \(\frac{1}{4a}\). Here, \(a = -1\), so the focus is located at \((-3, -\frac{5}{4}) = (-3, -1.25)\).
- **Axis of Symmetry:** The axis of symmetry is the vertical line through the vertex, \( x = -3 \).
- **Directrix:** The directrix is a horizontal line, and it is \( y = -1 + \frac{1}{4} = -\frac{3}{4} = -0.75 \).

### 2. \( x = -\frac{1}{4}(y + 2)^2 \)

This equation represents a parabola that opens to the left.

- **Vertex:** The vertex form is \( x = a(y-k)^2 + h \), where \((h, k)\) is the vertex. The vertex is \((-0.25, -2)\).
- **Focus:** The distance from the vertex to the focus is \(\frac{1}{4a}\). Here, \(a = -\frac{1}{4}\), so the focus is \((-0.5, -2)\).
- **Axis of Symmetry:** The axis of symmetry is the horizontal line through the vertex, \( y = -2 \).
- **Directrix:** The directrix is a vertical line, and it is \( x = -0.25 + 0.25 = 0 \).

### 3. \( -(1/3)(x-3) = (y + 5)^2 \)

This equation also represents a parabola that opens to the left.

- **Vertex:** The vertex form is \( (y-k)^2 = 4p(x-h) \). Rewrite the equation as \( (y + 5)^2 = -\frac{1}{3}(x - 3) \), so the vertex is \((3, -5)\).
- **Focus:** Here, \(4p = -\frac{1}{3}\), so \( p = -\frac{1}{12} \), and the focus is \((\frac{35}{12}, -5)\).
- **Axis of Symmetry:** The axis of symmetry is the horizontal line through the vertex, \( y = -5 \).
- **Directrix:** The directrix is a vertical line, \( x = 3 + \frac{1}{12} = 3.083 \).

### 4. \( -(y + 2) = (x - 2)^2 \)

This equation represents a parabola that opens downward.

- **Vertex:** The vertex is \((2, -2)\).
- **Focus:** The distance from the vertex to the focus is \( \frac{1}{4} \). So the focus is \((2, -2.25)\).
- **Axis of Symmetry:** The axis of symmetry is the vertical line \( x = 2 \).
- **Directrix:** The directrix is a horizontal line \( y = -1.75 \).

### 5. \( 3y + 4x = -(2x^2) - 14 \)

Rewrite the equation to fit the standard form:
\[ y = -\frac{2}{3}x^2 - \frac{4}{3}x - \frac{14}{3} \]

This is a parabola that opens downward.

- **Vertex:** To find the vertex, complete the square or use the vertex formula \( x = -\frac{b}{2a} \). After finding \( x \), substitute it back to get \( y \). The vertex here is \( \left(\frac{-4}{-4/3}, \frac{-4/3}{-14/3}\right)\).
- **Focus, Axis of Symmetry, and Directrix:** These are more complicated due to the involvement of more general quadratic functions.

### 6. \( x = -2(y + 2)^2 \)

This equation represents a parabola that opens to the left.

- **Vertex:** The vertex is \((-2, -2)\).
- **Focus:** The distance from the vertex to the focus is \(\frac{1}{4}(-2)\). So the focus is \((-2.25, -2)\).
- **Axis of Symmetry:** The axis of symmetry is the horizontal line \( y = -2 \).
- **Directrix:** The directrix is a vertical line \( x = -1.75 \).

### Graph Sketches

Each parabola can be sketched based on the direction they open (up, down, left, right), with the vertex as the central point. Then draw the focus point, and the directrix line on the opposite side.

**Do you want detailed steps on sketching or finding specific values for the above?**

### Next Steps:
1. Do you want to go over the process of completing the square for vertex form?
2. Would you like to calculate the exact focus and directrix for equation 5?
3. Do you need help with graphing these functions on paper or a tool?
4. Would you like to analyze any parabola in a specific context?
5. Do you want to understand how changing constants affect the shape and position?

**Tip:** Remember, parabolas always have their vertex as the midpoint between the focus and the directrix.

Learn how to identify the vertex, focus, axis of symmetry, and directrix of parabolas with detailed explanations and sketches. This problem covers key algebraic concepts related to parabolas and quadratic equations. Suitable for high school students.

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