https://file.aiquickdraw.com/mathbot/file/1723168515423ymy3y0r2.jpg

The equation given is \( f(x) = \frac{1}{2}|x + 3| - 4 \). Let's break down how to sketch the graph of this function.

### Step 1: Understand the Basic Shape
The function \( f(x) = \frac{1}{2}|x + 3| - 4 \) is based on the absolute value function \( |x| \), which typically forms a "V" shape.

### Step 2: Horizontal Shift
The expression inside the absolute value, \( x + 3 \), causes a horizontal shift to the left by 3 units. This means the vertex of the "V" shape will be at \( x = -3 \).

### Step 3: Vertical Stretch and Compression
The coefficient \( \frac{1}{2} \) before the absolute value indicates a vertical compression. Normally, the "V" shape would rise or fall with a slope of 1, but in this case, it will rise or fall with a slope of \( \frac{1}{2} \).

### Step 4: Vertical Shift
The \( -4 \) at the end shifts the entire graph downward by 4 units. This means the vertex of the "V" will now be at \( (-3, -4) \).

### Step 5: Sketching the Graph
1. **Vertex**: The vertex of the graph is at \( (-3, -4) \).
2. **Slopes**: 
   - To the right of \( x = -3 \), the graph will rise with a slope of \( \frac{1}{2} \).
   - To the left of \( x = -3 \), the graph will fall with a slope of \( -\frac{1}{2} \).
   
This information allows us to sketch the graph on the coordinate grid provided in the image.

Would you like more details or have any questions about this process?

### Additional Questions:
1. What is the effect of the absolute value on the graph of a function?
2. How do horizontal shifts affect the graph of a function?
3. What happens when you apply a vertical stretch or compression to a function?
4. How does adding or subtracting a constant outside the function affect its graph?
5. Can you describe how to determine the vertex of an absolute value function?
6. What would happen if the coefficient inside the absolute value was negative?
7. How would you graph \( f(x) = \frac{1}{2}|x| - 4 \) without the horizontal shift?
8. What is the general form of an absolute value function, and how do you interpret each component?

### Tip:
When sketching the graph of a transformed function, always start by identifying how each transformation (horizontal shift, vertical shift, stretch/compression) affects the basic graph of the function.

Learn how to sketch the graph of the function f(x) = 1/2|x + 3| - 4 with a detailed step-by-step guide. Understand the effect of absolute value, horizontal shifts, vertical stretch/compression, and vertical shifts on the graph. Suitable for high school students studying transformations of functions.

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