A.

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A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. Upper A parabola that opens upwards has vertex left parenthesis negative 3 comma negative 4 right parenthesis and passes through the points left parenthesis negative 4 comma negative 3 right parenthesis and left parenthesis negative 1 comma 0 right parenthesis .

B.

-1010-1010xy

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. Upper A parabola that opens downwards has vertex left parenthesis negative 3 comma negative 4 right parenthesis and passes through the points left parenthesis negative 4 comma negative 5 right parenthesis and left parenthesis negative 1 comma negative 8 right parenthesis .

C.

-1010-1010xy

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. Upper A parabola that opens downwards has vertex left parenthesis 3 comma negative 4 right parenthesis and passes through the points left parenthesis 2 comma negative 5 right parenthesis and left parenthesis 5 comma negative 8 right parenthesis .

D.

-1010-1010xy

Let's break down each of the cases to formulate the equations of the parabolas.

### General form of a parabola equation:
A parabola with a vertex \((h, k)\) can be expressed as:
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex, and \(a\) determines the direction and width of the parabola.

### Case A:
- **Vertex**: \((-3, -4)\)
- **Direction**: Opens upwards
- **Points**: Passes through \((-4, -3)\) and \((-1, 0)\)

1. **Equation form**: 
   \[
   y = a(x + 3)^2 - 4
   \]
   We use \((h, k) = (-3, -4)\).

2. **Find \(a\)** using the point \((-4, -3)\):
   \[
   -3 = a(-4 + 3)^2 - 4
   \]
   \[
   -3 = a(1)^2 - 4
   \]
   \[
   a = 1
   \]
   So the equation is:
   \[
   y = (x + 3)^2 - 4
   \]

### Case B:
- **Vertex**: \((-3, -4)\)
- **Direction**: Opens downwards
- **Points**: Passes through \((-4, -5)\) and \((-1, -8)\)

1. **Equation form**: 
   \[
   y = a(x + 3)^2 - 4
   \]
   Again, using the vertex \((-3, -4)\).

2. **Find \(a\)** using the point \((-4, -5)\):
   \[
   -5 = a(-4 + 3)^2 - 4
   \]
   \[
   -5 = a(1)^2 - 4
   \]
   \[
   a = -1
   \]
   The equation is:
   \[
   y = -(x + 3)^2 - 4
   \]

### Case C:
- **Vertex**: \((3, -4)\)
- **Direction**: Opens downwards
- **Points**: Passes through \((2, -5)\) and \((5, -8)\)

1. **Equation form**: 
   \[
   y = a(x - 3)^2 - 4
   \]
   We use \((h, k) = (3, -4)\).

2. **Find \(a\)** using the point \((2, -5)\):
   \[
   -5 = a(2 - 3)^2 - 4
   \]
   \[
   -5 = a(1)^2 - 4
   \]
   \[
   a = -1
   \]
   The equation is:
   \[
   y = -(x - 3)^2 - 4
   \]

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Would you like to go deeper into any of the cases? 

Here are 5 related questions:
1. How do you determine if a parabola opens upwards or downwards?
2. How do you find the vertex of a parabola from its equation?
3. What is the significance of the constant \(a\) in a parabola equation?
4. How can you use two points on a parabola to determine its equation?
5. What is the effect of shifting the vertex on the shape of the parabola?

**Tip**: The vertex form of a parabola \[y = a(x - h)^2 + k\] is very helpful in graphing and finding properties such as the vertex and direction of opening.

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A parabola that opens upwards has vertex (-3, -4) and passes through the points (-4, -3) and (-1, 0). Another parabola opens downwards with vertex (-3, -4) and passes through (-4, -5) and (-1, -8). Lastly, a third parabola opens downwards with vertex (3, -4) and passes through (2, -5) and (5, -8).

Explore how to determine the equations of parabolas given their vertex and points. This problem involves key algebraic concepts, including the vertex form of parabolas and solving for 'a'. Suitable for students in Grades 9-11.

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