At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Join our Q&A platform and get accurate answers to all your questions from professionals across multiple disciplines. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
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].
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.