Write an equation for the function whose graph is described.
The shape of 
f(x) = 

x
,
 but shifted eight units down and then reflected in both the x-axis and the y-axis
g(x) = 
 

To create the equation for the function described, let's break it down step by step:

1. **Original function**: The original function is \( f(x) = x \).
2. **Shifted 8 units down**: Shifting a function down by 8 units means subtracting 8 from the function. So, the new function becomes:
   \[
   f(x) = x - 8
   \]
3. **Reflection in the x-axis**: Reflecting a function over the x-axis means multiplying the entire function by \(-1\). This transforms the function into:
   \[
   f(x) = -(x - 8)
   \]
4. **Reflection in the y-axis**: Reflecting the function over the y-axis means replacing \(x\) with \(-x\). So, we replace \(x\) in the current function:
   \[
   g(x) = -(-x - 8)
   \]
   Simplifying this expression:
   \[
   g(x) = x + 8
   \]

Thus, the equation of the new function is:
\[
g(x) = x + 8
\]

Would you like more details or have any questions? Here are 5 related questions to deepen your understanding:

1. How does shifting a function up or down affect its graph?
2. What is the difference between reflecting a function over the x-axis and the y-axis?
3. Can you provide an example of a function reflected only over the y-axis?
4. How does combining transformations (shifts, reflections) affect the final function?
5. What is the impact of reflecting a quadratic function over both axes?

**Tip**: Always apply reflections carefully, and remember the order in which transformations are applied (reflections before shifts).

Write an equation for the function whose graph is described. The shape of f(x) = x, but shifted eight units down and then reflected in both the x-axis and the y-axis. g(x) = 

Learn how to transform the function f(x) = x by shifting it 8 units down and reflecting it over both the x-axis and y-axis. This step-by-step solution covers function transformations, including shifts and reflections, suitable for Grades 8-10 students.

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