https://file.aiquickdraw.com/mathbot/file/172909430475631u31iuh.jpg

The equation given in the image is:

\[
y = 3 |x - 3| + 4
\]

This represents a piecewise linear function involving an absolute value. Here's how to interpret and graph it:

### Key Features of the Equation:

1. **Vertex**: The absolute value function creates a V-shape, and the vertex of this function occurs where the expression inside the absolute value is zero, i.e., \(x - 3 = 0\). This gives \(x = 3\). 
   - Plugging \(x = 3\) into the equation, we get \(y = 3(0) + 4 = 4\). So, the vertex is at the point \((3, 4)\).
   
2. **Slope**: The coefficient 3 in front of the absolute value affects the slope of the lines on either side of the vertex:
   - For \(x > 3\), the equation behaves like \(y = 3(x - 3) + 4\), so the slope is 3.
   - For \(x < 3\), the equation behaves like \(y = 3(-x + 3) + 4\), so the slope is \(-3\).

3. **Vertical Shift**: The \(+4\) shifts the graph upwards by 4 units.

### Graphing the Function:
1. **Start at the vertex**: Plot the vertex at \((3, 4)\).
2. **Draw the right arm**: From the vertex, the slope on the right side of \(x = 3\) is 3. For every unit increase in \(x\), \(y\) increases by 3.
3. **Draw the left arm**: On the left side of \(x = 3\), the slope is \(-3\). For every unit decrease in \(x\), \(y\) decreases by 3.
4. **Include Symmetry**: The graph is symmetric around the vertical line \(x = 3\).

If you want to know how to construct the graph step-by-step or how to calculate specific points, let me know!

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Would you like further details or clarification on how to draw the graph? Here are five related questions to consider:

1. What does the absolute value function do to the shape of the graph?
2. How would the graph change if the slope (3) were different?
3. What happens to the vertex if the equation is modified to \(y = 3|x + 2| + 4\)?
4. Can we describe the behavior of the function for large values of \(x\)?
5. How does the transformation of \(y = |x|\) affect the graph's position?

**Tip**: When graphing absolute value functions, always start by finding the vertex (the point where the absolute value expression equals zero).

Graph the function y = 3 |x - 3| + 4 as shown in the image.

Learn how to graph the absolute value function y = 3|x - 3| + 4 with step-by-step instructions. Understand the vertex, slope, and shifts of the graph, and see how to apply these concepts to graph absolute value functions. Ideal for students in Grades 8-10.

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