Find g(x), where g(x) is the translation 5 units left and 5 units up of f(x)=|x|.
Write your answer in the form a|x–h|+k, where a, h, and k are integers.
g(x)=


To find the translation of \( f(x) = |x| \) that moves 5 units to the left and 5 units up, we need to apply transformations accordingly:

1. **Left Translation by 5 Units**: Shifting left means we add 5 to \( x \). So, we replace \( x \) with \( x + 5 \) in the function.
   - This gives \( |x + 5| \).

2. **Up Translation by 5 Units**: Shifting up means we add 5 to the entire function.
   - This gives \( |x + 5| + 5 \).

Thus, the translated function \( g(x) \) is:
\[
g(x) = |x + 5| + 5
\]

This is already in the form \( a|x - h| + k \), where:
- \( a = 1 \),
- \( h = -5 \),
- \( k = 5 \).

Would you like more details or have any questions?

Here are five related questions:
1. How do different translations (left, right, up, down) affect the graph of an absolute value function?
2. What would happen if we reflected the graph of \( f(x) = |x| \) across the x-axis?
3. How can we identify the vertex of a translated absolute value function?
4. How do we describe horizontal and vertical stretches or compressions of \( |x| \)?
5. What effect does changing the value of \( a \) have on the graph of \( a|x| \)?

**Tip:** Translating a function left or right involves changing the input (inside the absolute value), while translating up or down affects the output (outside the absolute value).

Find g(x), where g(x) is the translation 5 units left and 5 units up of f(x)=|x|. Write your answer in the form a|x–h|+k, where a, h, and k are integers. g(x)=

Explore how to translate the absolute value function f(x) = |x| by 5 units left and 5 units up. This guide covers the transformation of the function into the form g(x) = a|x - h| + k, suitable for students in Grades 9-12.

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