Question
What is the vertex form of the graphed equation?
(1 point)
Responses
y=−2(x+3)2+1
y is equal to negative 2 times open paren x plus 3 close paren squared plus 1
y=−2(x−3)2+1
y is equal to negative 2 times open paren x minus 3 close paren squared plus 1
y=2(x+3)2−1
y is equal to 2 times open paren x plus 3 close paren squared minus 1
y=−2(x+3)2−1
(1 point)
Responses
y=−2(x+3)2+1
y is equal to negative 2 times open paren x plus 3 close paren squared plus 1
y=−2(x−3)2+1
y is equal to negative 2 times open paren x minus 3 close paren squared plus 1
y=2(x+3)2−1
y is equal to 2 times open paren x plus 3 close paren squared minus 1
y=−2(x+3)2−1
Answers
GPT-4o mini
To determine the correct vertex form of the graphed equation, we should identify the vertex of the parabola, which is typically in the format \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. Additionally, the sign and magnitude of "a" indicate the direction and width of the parabola.
Since we don't see the actual graph here, let's analyze the given options:
1. **Option 1:** \( y = -2(x + 3)^2 + 1 \)
- Vertex at \( (-3, 1) \)
- Opens downwards since \( a = -2 \).
2. **Option 2:** \( y = -2(x - 3)^2 + 1 \)
- Vertex at \( (3, 1) \)
- Opens downwards since \( a = -2 \).
3. **Option 3:** \( y = 2(x + 3)^2 - 1 \)
- Vertex at \( (-3, -1) \)
- Opens upwards since \( a = 2 \).
4. **Option 4:** \( y = -2(x + 3)^2 - 1 \)
- Vertex at \( (-3, -1) \)
- Opens downwards since \( a = -2 \).
To identify which vertex form corresponds to the graph, look for a vertex point and whether the graph opens upward or downward.
If you could provide details regarding the vertex location and the direction of the parabola (opens up or down), I can help you choose the correct equation from the options provided.
Since we don't see the actual graph here, let's analyze the given options:
1. **Option 1:** \( y = -2(x + 3)^2 + 1 \)
- Vertex at \( (-3, 1) \)
- Opens downwards since \( a = -2 \).
2. **Option 2:** \( y = -2(x - 3)^2 + 1 \)
- Vertex at \( (3, 1) \)
- Opens downwards since \( a = -2 \).
3. **Option 3:** \( y = 2(x + 3)^2 - 1 \)
- Vertex at \( (-3, -1) \)
- Opens upwards since \( a = 2 \).
4. **Option 4:** \( y = -2(x + 3)^2 - 1 \)
- Vertex at \( (-3, -1) \)
- Opens downwards since \( a = -2 \).
To identify which vertex form corresponds to the graph, look for a vertex point and whether the graph opens upward or downward.
If you could provide details regarding the vertex location and the direction of the parabola (opens up or down), I can help you choose the correct equation from the options provided.
It starts at -8 and goes up to -3 then back to -8
GPT-4o mini
Based on your description, the graph has a minimum point (the vertex) at \( y = -3 \), and it returns to \( y = -8 \). This indicates that the parabola opens upwards with the vertex being the lowest point.
Given that the vertex is at \( (-3, -3) \) and it opens upwards, the vertex form of the parabola would match the option that has a positive leading coefficient (indicating upward direction).
Let's evaluate the options with this in mind:
1. **Option 1:** \( y = -2(x + 3)^2 + 1 \)
- Vertex at \( (-3, 1) \) (doesn't match).
2. **Option 2:** \( y = -2(x - 3)^2 + 1 \)
- Vertex at \( (3, 1) \) (doesn't match).
3. **Option 3:** \( y = 2(x + 3)^2 - 1 \)
- Vertex at \( (-3, -1) \) (doesn't match).
4. **Option 4:** \( y = -2(x + 3)^2 - 1 \)
- Vertex at \( (-3, -1) \) (doesn't match).
None of the options precisely match the vertex of \( (-3, -3) \). However, if we consider that the graph reaches a minimum at \( y = -3 \) while opening up, it seems there's no provided equation that fits exactly.
To maintain clarity, please ensure the vertex from your description is correct. If it truly is at \( (-3, -3) \) and opens upwards, it may be possible that there’s a miscommunication or an option not covered in your selections.
If you are looking for the general form from the vertex \( (-3, -3) \) that opens upwards, it would look like:
\[ y = a(x + 3)^2 - 3 \]
Where \( a > 0 \).
Given your selections, none directly corresponds, but if the vertex position or direction you provided differs, please clarify!
Given that the vertex is at \( (-3, -3) \) and it opens upwards, the vertex form of the parabola would match the option that has a positive leading coefficient (indicating upward direction).
Let's evaluate the options with this in mind:
1. **Option 1:** \( y = -2(x + 3)^2 + 1 \)
- Vertex at \( (-3, 1) \) (doesn't match).
2. **Option 2:** \( y = -2(x - 3)^2 + 1 \)
- Vertex at \( (3, 1) \) (doesn't match).
3. **Option 3:** \( y = 2(x + 3)^2 - 1 \)
- Vertex at \( (-3, -1) \) (doesn't match).
4. **Option 4:** \( y = -2(x + 3)^2 - 1 \)
- Vertex at \( (-3, -1) \) (doesn't match).
None of the options precisely match the vertex of \( (-3, -3) \). However, if we consider that the graph reaches a minimum at \( y = -3 \) while opening up, it seems there's no provided equation that fits exactly.
To maintain clarity, please ensure the vertex from your description is correct. If it truly is at \( (-3, -3) \) and opens upwards, it may be possible that there’s a miscommunication or an option not covered in your selections.
If you are looking for the general form from the vertex \( (-3, -3) \) that opens upwards, it would look like:
\[ y = a(x + 3)^2 - 3 \]
Where \( a > 0 \).
Given your selections, none directly corresponds, but if the vertex position or direction you provided differs, please clarify!
pick one
GPT-4o mini
Based on your description that the graph reaches a minimum at \(y = -3\) and goes back to \(y = -8\) (suggesting it opens upwards), the closest option in the list you provided would be:
**Option 3:** \(y = 2(x + 3)^2 - 1\)
However, please note that this option has its vertex at \( (-3, -1) \), which doesn't match the vertex point you mentioned. Given the options you provided, this is the best selection based on the graph opening upwards.
If the vertex is indeed at \( (-3, -3) \) as you suggested, none of the options reflect that accurately. But based on directionality (opening upwards) and vertex format, **Option 3** is the best choice among the provided alternatives.
**Option 3:** \(y = 2(x + 3)^2 - 1\)
However, please note that this option has its vertex at \( (-3, -1) \), which doesn't match the vertex point you mentioned. Given the options you provided, this is the best selection based on the graph opening upwards.
If the vertex is indeed at \( (-3, -3) \) as you suggested, none of the options reflect that accurately. But based on directionality (opening upwards) and vertex format, **Option 3** is the best choice among the provided alternatives.