Answer :
To determine which of the given equations represents a parabola with a vertex at [tex]\((0, -5)\)[/tex], we need to analyze each equation and identify the vertex of the parabola represented by it.
### Step-by-Step Analysis
1. Equation: [tex]\( y = x^2 + 5 \)[/tex]
- The general form of a parabola is [tex]\( y = a(x-h)^2 + k \)[/tex].
- By comparing [tex]\( y = x^2 + 5 \)[/tex] with [tex]\( y = a(x-h)^2 + k \)[/tex], we can see that [tex]\( h = 0 \)[/tex] and [tex]\( k = 5 \)[/tex].
- Therefore, the vertex of [tex]\( y = x^2 + 5 \)[/tex] is at [tex]\( (0, 5) \)[/tex].
- This does not match the given vertex [tex]\( (0, -5) \)[/tex].
2. Equation: [tex]\( y = x^2 - 5 \)[/tex]
- Again, comparing with [tex]\( y = a(x-h)^2 + k \)[/tex], we have [tex]\( h = 0 \)[/tex] and [tex]\( k = -5 \)[/tex].
- Therefore, the vertex of [tex]\( y = x^2 - 5 \)[/tex] is at [tex]\( (0, -5) \)[/tex].
- This matches the given vertex [tex]\( (0, -5) \)[/tex].
3. Equation: [tex]\( y = (x-5)^2 \)[/tex]
- Here, by comparing with [tex]\( y = a(x-h)^2 + k \)[/tex], we have [tex]\( h = 5 \)[/tex] and [tex]\( k = 0 \)[/tex].
- Therefore, the vertex of [tex]\( y = (x-5)^2 \)[/tex] is at [tex]\( (5, 0) \)[/tex].
- This does not match the given vertex [tex]\( (0, -5) \)[/tex].
4. Equation: [tex]\( y = (x+5)^2 \)[/tex]
- Comparing with [tex]\( y = a(x-h)^2 + k \)[/tex], we find [tex]\( h = -5 \)[/tex] and [tex]\( k = 0 \)[/tex].
- Therefore, the vertex of [tex]\( y = (x+5)^2 \)[/tex] is at [tex]\( (-5, 0) \)[/tex].
- This does not match the given vertex [tex]\( (0, -5) \)[/tex].
### Conclusion
After analyzing each equation, we can see that the equation which represents the parabola with a vertex at [tex]\((0, -5)\)[/tex] is:
[tex]\[ y = x^2 - 5 \][/tex]
So, the correct answer is [tex]\( \boxed{2} \)[/tex].
### Step-by-Step Analysis
1. Equation: [tex]\( y = x^2 + 5 \)[/tex]
- The general form of a parabola is [tex]\( y = a(x-h)^2 + k \)[/tex].
- By comparing [tex]\( y = x^2 + 5 \)[/tex] with [tex]\( y = a(x-h)^2 + k \)[/tex], we can see that [tex]\( h = 0 \)[/tex] and [tex]\( k = 5 \)[/tex].
- Therefore, the vertex of [tex]\( y = x^2 + 5 \)[/tex] is at [tex]\( (0, 5) \)[/tex].
- This does not match the given vertex [tex]\( (0, -5) \)[/tex].
2. Equation: [tex]\( y = x^2 - 5 \)[/tex]
- Again, comparing with [tex]\( y = a(x-h)^2 + k \)[/tex], we have [tex]\( h = 0 \)[/tex] and [tex]\( k = -5 \)[/tex].
- Therefore, the vertex of [tex]\( y = x^2 - 5 \)[/tex] is at [tex]\( (0, -5) \)[/tex].
- This matches the given vertex [tex]\( (0, -5) \)[/tex].
3. Equation: [tex]\( y = (x-5)^2 \)[/tex]
- Here, by comparing with [tex]\( y = a(x-h)^2 + k \)[/tex], we have [tex]\( h = 5 \)[/tex] and [tex]\( k = 0 \)[/tex].
- Therefore, the vertex of [tex]\( y = (x-5)^2 \)[/tex] is at [tex]\( (5, 0) \)[/tex].
- This does not match the given vertex [tex]\( (0, -5) \)[/tex].
4. Equation: [tex]\( y = (x+5)^2 \)[/tex]
- Comparing with [tex]\( y = a(x-h)^2 + k \)[/tex], we find [tex]\( h = -5 \)[/tex] and [tex]\( k = 0 \)[/tex].
- Therefore, the vertex of [tex]\( y = (x+5)^2 \)[/tex] is at [tex]\( (-5, 0) \)[/tex].
- This does not match the given vertex [tex]\( (0, -5) \)[/tex].
### Conclusion
After analyzing each equation, we can see that the equation which represents the parabola with a vertex at [tex]\((0, -5)\)[/tex] is:
[tex]\[ y = x^2 - 5 \][/tex]
So, the correct answer is [tex]\( \boxed{2} \)[/tex].
