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!