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.