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The graph above is a transformation of the function f(x)=|x|

Write an equation for the function graphed above

The given graph is a transformation of the absolute value function \( f(x) = |x| \). Let's analyze the graph step by step to find the equation for \( g(x) \).

1. **Reflection**: The graph opens downward for \( x < 0 \), indicating a reflection over the x-axis on the left side of the y-axis. This means the graph for negative values of \( x \) is multiplied by \( -1 \).

2. **Slope**: For \( x \geq 0 \), the graph increases with a slope of 1, which is consistent with the parent function \( f(x) = |x| \). However, for \( x < 0 \), the slope of the graph is steeper. It has a slope of \( -2 \), indicating a vertical stretch by a factor of 2 for negative \( x \)-values.

3. **Vertex**: The vertex of the graph is at \( (0, -1) \). This indicates a downward vertical shift of 1 unit.

### Putting it all together:

The graph is a piecewise function with the following transformations:
- For \( x \geq 0 \), the function follows the standard absolute value function, but with a downward shift by 1 unit: \( x - 1 \).
- For \( x < 0 \), the graph has a vertical stretch by 2, and it is reflected (negative slope) with the same downward shift.

Thus, the equation for \( g(x) \) is:
\[
g(x) =
\begin{cases} 
-2x - 1 & \text{for } x < 0 \\
x - 1 & \text{for } x \geq 0
\end{cases}
\]

Would you like further explanation on any part of this process? 

Here are 5 related questions:
1. What transformations affect the shape of an absolute value graph?
2. How do you recognize vertical and horizontal shifts in a graph?
3. What does a vertical stretch mean in terms of transformations?
4. How can you write a piecewise function based on graph analysis?
5. How does reflecting a graph over the x-axis or y-axis affect its equation?

**Tip**: Always check for shifts, reflections, and stretches/compressions when identifying transformations of a function graph.

The graph above is a transformation of the function f(x) = |x|. Write an equation for the function graphed above.

Learn how to find the equation of a transformed absolute value function by analyzing the graph. This example involves identifying reflections, vertical stretches, and shifts to write a piecewise function. Suitable for high school students in Grades 9-12.

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