https://file.aiquickdraw.com/mathbot/file/1725646878574pvjm35in.jpg

The equation given is:

\[
f(x) = -2|x - 4| + 2
\]

This is an absolute value function of the form \( f(x) = a|x - h| + k \), where:
- \( a = -2 \), which controls the vertical stretch or compression and the reflection over the x-axis.
- \( h = 4 \), which is the horizontal shift, moving the vertex to \( x = 4 \).
- \( k = 2 \), which is the vertical shift, moving the vertex up by 2 units.

### Key points to graph this function:
1. **Vertex**: The vertex is located at \( (h, k) = (4, 2) \).
2. **Slope**: Since \( a = -2 \), the slope of the lines is \( -2 \) on both sides of the vertex. On the left side (for \( x < 4 \)), the function will have a positive slope (rising), and on the right side (for \( x > 4 \)), it will have a negative slope (falling).
3. **Reflection**: The negative sign in front of the 2 means the graph will open downward.

### Steps for graphing:
1. **Plot the vertex at (4, 2)**.
2. **Plot additional points**:
   - For \( x = 3 \): \( f(3) = -2|3 - 4| + 2 = -2(1) + 2 = 0 \). So the point is \( (3, 0) \).
   - For \( x = 5 \): \( f(5) = -2|5 - 4| + 2 = -2(1) + 2 = 0 \). So the point is \( (5, 0) \).
   - For \( x = 2 \): \( f(2) = -2|2 - 4| + 2 = -2(2) + 2 = -4 + 2 = -2 \). So the point is \( (2, -2) \).
   - For \( x = 6 \): \( f(6) = -2|6 - 4| + 2 = -2(2) + 2 = -4 + 2 = -2 \). So the point is \( (6, -2) \).

3. **Draw the graph**: Connect the points with two straight lines that form a "V" shape, opening downward, centered at the vertex \( (4, 2) \).

Would you like more details on this graph, or do you have any questions?

### Related questions:
1. How does changing the value of \( a \) affect the shape of the absolute value graph?
2. What happens if we change the value of \( h \) in the equation \( f(x) = -2|x - h| + 2 \)?
3. How would the graph change if the constant \( k \) were increased or decreased?
4. How can we find the x-intercepts of the graph?
5. What does the reflection of the graph tell us about the function's behavior?

### Tip:
When graphing absolute value functions, always locate the vertex first, and then use the slope to plot points symmetrically on both sides of the vertex.

Learn how to graph the absolute value function f(x) = -2|x - 4| + 2. This problem covers key algebraic concepts like vertex identification, vertical shifts, and graphing techniques for absolute value functions. Suitable for Grades 8-10.

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