https://file.aiquickdraw.com/mathbot/file/1726718761213yod58euv.jpg

Use the graph of a transformed toolkit function to write a formula for the resulting function.
y =

From the graph you provided, we can infer that this is a transformation of the **absolute value function** \( f(x) = |x| \). The absolute value function typically has a V-shape, and it seems like this graph has been reflected, translated, and possibly scaled.

### Observations:
1. **Vertex**: The vertex of the graph is located at \( (0, 5) \), which indicates a vertical translation of the function up by 5 units.
2. **Reflection**: The graph opens downward, suggesting that the function has been reflected over the x-axis, meaning there is a negative coefficient.
3. **Slope**: The slopes of the lines on either side of the vertex are -1 and 1, which implies no horizontal scaling.

### Formula:
The general form of an absolute value function is:
\[
y = a|x - h| + k
\]
Where:
- \( a \) is the vertical stretch or compression and reflects the graph if \( a \) is negative,
- \( h \) and \( k \) represent horizontal and vertical shifts, respectively.

For this specific graph:
- \( a = -1 \) (reflection over the x-axis),
- \( h = 0 \) (no horizontal shift),
- \( k = 5 \) (vertical shift up by 5).

Thus, the equation of the function is:
\[
y = -|x| + 5
\]

Would you like further details on any transformations or other aspects of this function?

Here are 5 related questions for further exploration:
1. How do you determine the vertex of a transformed absolute value function?
2. What role does the coefficient \( a \) play in reflecting or scaling the graph?
3. How can we identify vertical or horizontal shifts from a graph?
4. How would the function change if the vertex were moved to a different point?
5. Can you describe how the domain and range of the function are affected by the transformations?

**Tip**: When analyzing transformations of basic functions, always check for vertical and horizontal shifts, reflections, and changes in slope (for linear portions) to find the equation.

Use the graph of a transformed toolkit function to write a formula for the resulting function. y = 

Analyze the transformed absolute value function y = -|x| + 5, involving reflections and vertical shifts. This problem explores how graph transformations affect the function's shape and position, suitable for Grades 8-10 students.

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