https://file.aiquickdraw.com/mathbot/file/17264553887808tkqvad5.jpg

The image shows the graph of a transformation of the function \( f(x) = x^2 \), with a new function \( g(x) \). Based on the graph, it seems that the function \( g(x) \) represents a horizontal shift of the original function \( f(x) = x^2 \).

Here’s the breakdown:
1. The vertex of \( f(x) = x^2 \) is originally at \( (0, 0) \).
2. In the graph provided, the vertex of \( g(x) \) appears to be at \( (3, 4) \).

This suggests a horizontal shift 3 units to the right and a vertical shift 4 units upwards. Therefore, the transformation applied to \( f(x) \) can be written as:

\[
g(x) = (x - 3)^2 + 4
\]

This equation represents the function that has been shifted horizontally and vertically.

Would you like a detailed explanation of how these transformations work?

Here are 5 related questions to expand your understanding:
1. What happens to the graph of \( f(x) = x^2 \) if you shift it left by 2 units?
2. How does a vertical stretch affect the function \( f(x) = x^2 \)?
3. What is the equation of \( f(x) = x^2 \) if the graph is reflected over the x-axis?
4. How do you find the vertex of a transformed quadratic function?
5. What is the effect of adding a coefficient in front of \( (x - 3)^2 \) on the graph?

**Tip:** When dealing with transformations, horizontal shifts involve changes inside the parentheses (i.e., \( (x - h) \)), while vertical shifts affect the constant outside (i.e., \( + k \)).

Below is a graph of g(x), a transformation of f(x) = x^2. What is the equation for g(x)?

Learn how to find the equation of g(x), a transformation of the quadratic function f(x) = x^2, by analyzing shifts in the vertex. This problem involves understanding horizontal and vertical shifts in graphs. Suitable for students in Grades 9-10.

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