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The problem asks for the function \( g(x) \), which is a translation of the function \( f(x) = x^2 \) by 8 units to the left and 9 units up.

To translate a function, we use the following rules:
1. To move the function left by \( h \) units, replace \( x \) with \( (x + h) \) in the function.
2. To move the function up by \( k \) units, add \( k \) to the function.

Given:
- Translation 8 units left: \( h = 8 \), so replace \( x \) with \( (x + 8) \).
- Translation 9 units up: \( k = 9 \), so add 9 to the function.

Therefore, starting with \( f(x) = x^2 \), the transformation gives:

\[
g(x) = (x + 8)^2 + 9
\]

This is the desired form of the function \( g(x) \).

Would you like any further clarification or details on this? Here are some related questions to expand on the concept:

1. How would you translate \( f(x) = x^2 \) by 5 units to the right and 3 units down?
2. What effect does translating a function left or right have on the graph of a quadratic function?
3. Can you describe the effect of translating a function upward or downward on its graph?
4. What is the vertex of \( g(x) = (x + 8)^2 + 9 \)?
5. How do you reflect a function like \( f(x) = x^2 \) over the x-axis?

**Tip**: Remember that translating a function horizontally affects the input variable (inside the parentheses), while vertical translations affect the entire function (outside the parentheses).

Find g(x), where g(x) is the translation 8 units left and 9 units up of f(x) = x^2. Write your answer in the form a(x−h)^2 + k, where a, h, and k are integers.

Learn how to translate the quadratic function f(x) = x^2 by 8 units left and 9 units up. This problem covers function transformation concepts and is suitable for students in Grades 9-11.

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