The graphs below shows g(x) which is a translation of f(x)=x^2 write the function rutfor g(x) write answer in the form a(x-h)^2 +k

To write the function \( g(x) \) in the form \( a(x-h)^2 + k \), we need to identify the vertex \((h, k)\) of each graph and any vertical scaling factor \( a \).

Graph 1

The vertex appears to be at \((0, 6)\), and the points \( (-2, 10) \) and \( (2, 10) \) suggest that this is a parabola that opens upwards.

To find \( a \):

• Substitute one of the points (e.g., \( (2, 10) \)) into the vertex form equation. \[ 10 = a(2 - 0)^2 + 6 \] \[ 10 = 4a + 6 \] \[ 4a = 4 \implies a = 1 \]

Thus, \[ g(x) = 1(x-0)^2 + 6 = (x)^2 + 6 \]

Final form: \[ g(x) = (x - 0)^2 + 6 \]

Graph 2

The vertex appears to be at \((-6, 0)\) as this point is the lowest point on the graph. The points \((-8, 4)\) and \((-4, 4)\) confirm that the parabola opens upwards.

To find \( a \):

• Substitute one of the points (e.g., \( (-8, 4) \)): \[ 4 = a(-8 + 6)^2 + 0 \] \[ 4 = a(-2)^2 \] \[ 4 = 4a \implies a = 1 \]

Thus, \[ g(x) = 1(x + 6)^2 + 0 = (x + 6)^2 \]

Final form: \[ g(x) = (x + 6)^2 \]

Graph 3

The vertex appears to be at \((7, 0)\). The points \( (5, 4) \) and \( (9, 4) \) confirm that the parabola opens upwards.

To find \( a \):

• Substitute one of the points (e.g., \( (5, 4) \)): \[ 4 = a(5 - 7)^2 + 0 \] \[ 4 = a(-2)^2 \] \[ 4 = 4a \implies a = 1 \]

Thus, \[ g(x) = 1(x - 7)^2 + 0 = (x - 7)^2 \]

Final form: \[ g(x) = (x - 7)^2 \]

Graph 4

The vertex appears to be at \((2, 1)\). The points \( (0, 5) \) and \( (4, 5) \) confirm that the parabola opens upwards.

To find \( a \):

• Substitute one of the points (e.g., \( (0, 5) \)): \[ 5 = a(0 - 2)^2 + 1 \] \[ 5 = 4a + 1 \] \[ 4 = 4a \implies a = 1 \]

Thus, \[ g(x) = 1(x - 2)^2 + 1 = (x - 2)^2 + 1 \]

Final form: \[ g(x) = (x - 2)^2 + 1 \]

Summary

1. Graph 1: \( g(x) = (x - 0)^2 + 6 \)
2. Graph 2: \( g(x) = (x + 6)^2 \)
3. Graph 3: \( g(x) = (x - 7)^2 \)
4. Graph 4: \( g(x) = (x - 2)^2 + 1 \)

Let's analyze Graph 1 again based on the points provided: (0, 6), (-2, 10), and (2, 10).

Analysis of Graph 1

Given:

• Vertex: The graph opens upwards and the lowest point (the vertex) is located at the midpoint of the two points \( (-2, 10) \) and \( (2, 10) \). The x-coordinate of the vertex is \( 0 \) because it's symmetrically placed between those two points. Thus, the vertex is \( (0, 6) \).

To find \( a \):

1. The vertex form is \( g(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. Here, \( h = 0 \) and \( k = 6 \), so \( g(x) = a(x - 0)^2 + 6 = ax^2 + 6 \).

2. Now, we will use one of the points to find \( a \). Let's use the point \( (2, 10) \): \[ 10 = a(2 - 0)^2 + 6 \] \[ 10 = 4a + 6 \] \[ 4 = 4a \implies a = 1 \]

Thus, the correct equation for Graph 1 is: \[ g(x) = (x - 0)^2 + 6 \quad \text{or simply} \quad g(x) = x^2 + 6 \]

Final answer for Graph 1: \[ g(x) = (x - 0)^2 + 6 \]

I'll assume the previous answer for Graph 1 was aligned with this conclusion, but if there is further clarification needed or if additional points are provided for context, please let me know!

Additionally, if you could specify what aspect of the answer you believe is incorrect, I'd be happy to adjust the analysis accordingly!

Thank you for providing more details about Graph 1. Based on the updated points: \( (0, 6), (-1, 7), (1, 7), (-2, 10), (2, 10) \), we can analyze the function more accurately.

Step-by-step Analysis of Graph 1

1. Identifying the Vertex:

   • The lowest point of the parabola is at \( (0, 6) \), which indicates that the vertex \( (h, k) \) is \( (0, 6) \).

2. Finding the Value of \( a \):

   • The points \( (-1, 7) \) and \( (1, 7) \) confirm that both these points are equidistant from the vertex and have the same y-value (7). We can use one of these points to find the value of \( a \).
   • Using the point \( (1, 7) \): \[ 7 = a(1 - 0)^2 + 6 \] \[ 7 = a(1) + 6 \] \[ 7 - 6 = a \implies a = 1 \]

With the vertex and the value of \( a \) determined:

• The function \( g(x) \) can now be written in vertex form: \[ g(x) = 1(x - 0)^2 + 6 = (x - 0)^2 + 6 = x^2 + 6 \]

Final Answer for Graph 1

\[ g(x) = (x - 0)^2 + 6 \]

Summary for Graphs

Let’s summarize the derived equations for all provided graphs:

1. Graph 1: \( g(x) = (x - 0)^2 + 6 \)
2. Graph 2: \( g(x) = (x + 6)^2 \) (from previous solution)
3. Graph 3: \( g(x) = (x - 7)^2 \) (from previous solution)
4. Graph 4: \( g(x) = (x - 2)^2 + 1 \) (from previous solution)

If you have any further corrections or additional graphs, feel free to share!