https://file.aiquickdraw.com/mathbot/file/1727051609735yyd3px0o.jpg

State the transformations from the parent graph then use the transformations to sketch the graph and state domain and range

The function given in the image is:

\[
f(x) = \frac{1}{3} \cdot |x - 7| - 4
\]

### Step 1: Identify the parent function
The parent function here is the absolute value function, \( f(x) = |x| \). This is a V-shaped graph with a vertex at the origin (0, 0).

### Step 2: Transformations
Now, let's break down the transformations applied to the parent function:

1. **Horizontal Shift ( \( x - 7 \) )**:  
   - The graph shifts **7 units to the right**. This is because the expression \( |x - 7| \) indicates a horizontal translation of 7 units to the right.

2. **Vertical Stretch by \( \frac{1}{3} \)**:  
   - The coefficient \( \frac{1}{3} \) in front of the absolute value function stretches the graph vertically by a factor of \( \frac{1}{3} \). This makes the V-shape of the graph wider (less steep).

3. **Vertical Shift ( -4 )**:  
   - The subtraction of 4 from the entire function shifts the graph **4 units downward**. So the vertex of the V-shape is lowered by 4 units.

### Step 3: Vertex of the transformed function
The vertex of the parent function \( f(x) = |x| \) is at (0, 0). After applying the transformations:
- Horizontal shift 7 units to the right → the vertex moves to \( (7, 0) \).
- Vertical shift 4 units down → the vertex moves to \( (7, -4) \).

So, the vertex of the transformed graph is at \( (7, -4) \).

### Step 4: Sketching the graph
- The graph has a V-shape, with the vertex at \( (7, -4) \).
- It opens upward, but is less steep than the parent graph due to the vertical stretch by \( \frac{1}{3} \).
- The slope of the "arms" of the V is \( \pm \frac{1}{3} \), meaning the graph rises/falls less steeply than the standard absolute value function.

### Step 5: Domain and Range
- **Domain**: The domain of the absolute value function is all real numbers, as the function is defined for any real \( x \).  
  Therefore, the domain is:
  \[
  \text{Domain}: (-\infty, \infty)
  \]
- **Range**: The lowest point of the graph is at \( y = -4 \), and the function increases without bound from there.  
  Therefore, the range is:
  \[
  \text{Range}: [-4, \infty)
  \]

Would you like more details or have any questions?

---

Here are five related questions for further exploration:
1. How would the graph change if the coefficient \( \frac{1}{3} \) were negative?
2. What effect does changing the \( -7 \) inside the absolute value to \( +7 \) have on the graph?
3. How would the range change if the vertical shift was \( +4 \) instead of \( -4 \)?
4. How does a horizontal stretch or compression differ from a vertical one in terms of transformation?
5. Can you express this function in piecewise form, and how does that help with graphing?

**Tip**: Always analyze shifts inside the absolute value for horizontal changes and constants outside for vertical changes!

State the transformations from the parent graph then use the transformations to sketch the graph and state domain and range.

Learn how to graph the absolute value function f(x) = (1/3) * |x - 7| - 4 by identifying and applying the correct transformations, including horizontal shifts, vertical stretches, and vertical shifts. Suitable for students in Grades 9-12.

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