The graph shown in your image corresponds to the function $y = \sqrt[3]{x} - 5$. Here's how the transformation is applied:

1. Parent Function: The parent function is $y = \sqrt[3]{x}$, which is the cube root of $x$. The graph of $y = \sqrt[3]{x}$ passes through the origin (0,0) and has a general S-shape, increasing for positive $x$ and decreasing for negative $x$.

2. Transformation: The given equation $y = \sqrt[3]{x} - 5$ involves a vertical shift. The "-5" indicates that the graph of $y = \sqrt[3]{x}$ has been shifted downward by 5 units. This means that each point on the graph of the parent function is lowered by 5 units on the y-axis.

3. Key Points:

   • The origin $(0, 0)$ in the parent function moves to $(0, -5)$.
   • Other points, such as $(1, 1)$ on the parent function, move to $(1, -4)$, and similarly for points on the negative side.

The graph shows this transformation accurately, with key points plotted at appropriate locations after the shift.

Would you like further details or explanations on this transformation?

Here are 5 related questions to deepen your understanding:

1. How would the graph change if the equation was $y = \sqrt[3]{x} + 5$?
2. What is the domain and range of the function $y = \sqrt[3]{x} - 5$?
3. How would the graph change if there was a horizontal shift applied (e.g., $y = \sqrt[3]{x-2} - 5$)?
4. What is the effect of a vertical stretch (e.g., $y = 2\sqrt[3]{x} - 5$) on the graph?
5. Can you describe the symmetry properties of the cube root function and how they apply after transformation?

Tip: Always look for vertical or horizontal shifts in the equation to easily identify how the graph moves relative to the parent function!