Question
Which response correctly writes the equation in vertex form for the function?
1. y=1/2(x+2)^2
2. y=1/2(x-2)^2
1. y=1/2(x+2)^2
2. y=1/2(x-2)^2
Answers
GPT-4o mini
To determine which response correctly writes the equation in vertex form, we first need to understand what vertex form is. The vertex form of a quadratic function is given by:
\[ y = a(x - h)^2 + k \]
where \( (h, k) \) is the vertex of the parabola.
Now, let's analyze the options provided:
1. \( y = \frac{1}{2}(x + 2)^2 \)
In this form, we can rewrite it as:
\[ y = \frac{1}{2}(x - (-2))^2 + 0 \]
This indicates that the vertex is at \( (-2, 0) \).
2. \( y = \frac{1}{2}(x - 2)^2 \)
This is already in vertex form, and we can see that the vertex is at \( (2, 0) \).
Both choices are valid vertex form equations, just for different parabola orientations and vertex locations.
If you need the one that represents a specific function or a specific vertex, you would choose appropriately based on those conditions. However, as both options are correct vertex forms, there isn't a singular correct answer without additional context.
\[ y = a(x - h)^2 + k \]
where \( (h, k) \) is the vertex of the parabola.
Now, let's analyze the options provided:
1. \( y = \frac{1}{2}(x + 2)^2 \)
In this form, we can rewrite it as:
\[ y = \frac{1}{2}(x - (-2))^2 + 0 \]
This indicates that the vertex is at \( (-2, 0) \).
2. \( y = \frac{1}{2}(x - 2)^2 \)
This is already in vertex form, and we can see that the vertex is at \( (2, 0) \).
Both choices are valid vertex form equations, just for different parabola orientations and vertex locations.
If you need the one that represents a specific function or a specific vertex, you would choose appropriately based on those conditions. However, as both options are correct vertex forms, there isn't a singular correct answer without additional context.