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The graphs of *f* and *g* are given. Find a formula for the function *g*.
*g*(*x*) =

To find the formula for the function \(g(x)\) based on the graph provided, we will analyze the transformations applied to the graph of \(f(x) = 4x^3\).

### Step-by-Step Analysis

1. **Identify the Given Functions:**
   - The blue graph is labeled \(f(x) = 4x^3\), which is a cubic function with a standard cubic shape. It passes through the origin \((0,0)\) and has an increasing trend.
   - The red graph represents the function \(g(x)\), which appears to be another cubic function, similar in shape but horizontally shifted.

2. **Observe the Properties of \(g(x)\):**
   - The graph of \(g(x)\) appears to be a horizontal shift of the graph of \(f(x) = 4x^3\).
   - By inspecting the graph, we see that the graph of \(g(x)\) is shifted 1 unit to the right compared to the graph of \(f(x)\).

3. **Determine the Transformation:**
   - A horizontal shift to the right by 1 unit means the function \(f(x)\) has been transformed into \(g(x) = f(x - 1)\).
   - Since \(f(x) = 4x^3\), substituting \(x - 1\) into \(f(x)\) gives us the equation for \(g(x)\).

4. **Formulate the Equation:**
   - The function \(g(x)\) is therefore:
   \[
   g(x) = 4(x - 1)^3
   \]

### Conclusion

The formula for the function \(g(x)\) is:

\[
g(x) = 4(x - 1)^3
\]

Would you like more details or have any questions?

### Related Questions:

1. What would the graph look like if the function \(g(x)\) was shifted 2 units to the left instead?
2. How does a vertical stretch or compression affect the shape of a cubic function?
3. What transformations affect the orientation (i.e., reflection) of a cubic graph?
4. How can we determine the inflection point of a cubic function from its graph?
5. What impact does changing the coefficient in front of \(x^3\) have on the graph's steepness?

### Tip:

When analyzing function transformations, look for shifts (horizontal or vertical), reflections, and stretches or compressions. Understanding how these affect the base function will help you quickly determine the new function's equation.

Discover how to find the formula for g(x) from given f(x) graphs using graph transformations and cubic function properties. This detailed explanation covers horizontal shifts and cubic function transformations, suitable for Grades 10-12 students.

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