https://file.aiquickdraw.com/mathbot/file/17289327052267rlk6l6i.jpg

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 \( (1, -5) \).

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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.

Use the graph of f(x) = x^3 to write an equation for the function represented by each graph.

Learn how to write equations for transformed cubic functions based on graphs. This lesson covers key concepts like horizontal and vertical shifts, reflections, and how to apply transformations to the cubic function f(x) = x^3. Suitable for high school students.

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