The image you provided contains two graphs, labeled (a) and (b), which are based on transformations of the cubic function $f(x) = x^3$. You are asked to write equations for the functions represented by each graph.

(a) The graph:

• The base function is $f(x) = x^3$, which is a standard cubic function.
• The point marked on the graph is $(2, 1)$, and it seems to have undergone a horizontal shift.
• The general form for the transformation of $f(x) = x^3$ is $g(x) = (x - h)^3 + k$, where $h$ is the horizontal shift, and $k$ is the vertical shift.

The point $(2, 1)$ suggests the function has been shifted horizontally to the right by 2 units, which gives the form: $g(x) = (x - 2)^3 + 1.$

However, there is an equation provided as $g(x) = (x - 3)^3$. This indicates the actual shift for this graph might be more complicated than what is suggested by the graph at first glance, or it was provided to match a specific value.

(b) The graph:

• Again, the base function is $f(x) = x^3$, but this time the function has been reflected and shifted.
• The point marked is $(1, -5)$, suggesting both horizontal and vertical shifts, and a reflection across the x-axis.
• The transformation of a cubic function that is reflected is of the form $g(x) = -f(x - h) + k$, where $h$ and $k$ indicate shifts.
• From the graph, there is a reflection in the x-axis and the point of interest is $(1, -5)$. Thus, the equation should be: $g(x) = -(x - 1)^3 - 5.$

This equation accounts for both the reflection and the shifts indicated by the point $(1, -5)$.

Do you want further details on how the transformations work or have any questions about the steps? Here are some questions to deepen your understanding:

1. How do we determine vertical and horizontal shifts in cubic functions?
2. What impact does reflecting a function have on its graph?
3. Why is the equation for graph (a) given as $g(x) = (x - 3)^3$ instead of something different?
4. How would you describe the end behavior of cubic functions?
5. How do you recognize if a function has been reflected from its graph?

Tip:

When analyzing transformations, always check how the key points (like the inflection point) of the base graph are shifted to determine the horizontal and vertical changes.