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The graph shows g(x), which is a translation of f(x)=x2. Write the function rule for g(x).
Write your answer in the form a(x–h)2+k, where a, h, and k are integers or simplified fractions.
g(x)=

The graph provided shows a parabola, which is a transformation of the basic quadratic function \( f(x) = x^2 \). Based on the image, I will determine the values of \( a \), \( h \), and \( k \) to write the function \( g(x) \) in the form:

\[
g(x) = a(x - h)^2 + k
\]

### Observations from the graph:
1. **Vertex**: The vertex of the parabola is located at \( (4, 2) \), meaning the graph has been translated 4 units to the right and 2 units up. Therefore, \( h = 4 \) and \( k = 2 \).
   
2. **Shape and Orientation**: The parabola opens upwards, and based on its width, there doesn't appear to be any vertical stretching or compression compared to \( f(x) = x^2 \). Thus, \( a = 1 \).

### Function Rule:
Given these transformations, the equation for \( g(x) \) is:

\[
g(x) = (x - 4)^2 + 2
\]

This is the rule for the function \( g(x) \), describing the parabola in the graph.

Would you like further details or have any other questions?

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Here are 5 follow-up questions for exploration:
1. How would the function change if the parabola were reflected across the x-axis?
2. What happens to the graph if the value of \( a \) is greater than 1?
3. How would you describe the effect of \( h \) on the graph of a parabola?
4. If \( k \) were negative, how would that affect the graph?
5. Can you identify the domain and range of the function \( g(x) \)?

**Tip**: When analyzing the transformation of a function, always start by identifying the vertex to help you determine the shifts in the graph.

The graph shows g(x), which is a translation of f(x)=x^2. Write the function rule for g(x). Write your answer in the form a(x–h)^2+k, where a, h, and k are integers or simplified fractions. g(x)=

Learn how to determine the function rule for g(x), a transformation of the quadratic function f(x) = x^2. Explore the process of identifying vertex shifts and transformations, suitable for students in Grades 8-10.

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