Starting with the graph of a basic​ function, graph the following function using the techniques of​ shifting, compressing,​ stretching, and/or reflecting. Find the domain and range of the function. 
g left parenthesis x right parenthesis equals left parenthesis x plus 1 right parenthesis cubed plus 6
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Part 1
Use the graphing tool to graph the equation.



The given function is \( g(x) = (x + 1)^3 + 6 \). We need to analyze how this function can be graphed using techniques like shifting, compressing, stretching, and reflecting, and then find the domain and range.

### Step-by-Step Graphing Process:

1. **Start with the basic function**: 
   The basic function is \( f(x) = x^3 \). This is a cubic function with its characteristic shape (an "S"-like curve that passes through the origin). The domain of \( f(x) = x^3 \) is \( (-\infty, \infty) \), and its range is also \( (-\infty, \infty) \).

2. **Horizontal Shift**:
   The function has been modified to \( g(x) = (x + 1)^3 + 6 \). The \( x + 1 \) inside the cube indicates a **horizontal shift**. Specifically, the graph of \( f(x) = x^3 \) is shifted **1 unit to the left**. This means that instead of the curve passing through \( (0,0) \), it now passes through \( (-1, 0) \).

3. **Vertical Shift**:
   The \( +6 \) outside the cubic term indicates a **vertical shift**. The graph is shifted **6 units upward**, so the point \( (-1, 0) \) moves to \( (-1, 6) \).

Thus, the graph of \( g(x) = (x + 1)^3 + 6 \) is the graph of \( f(x) = x^3 \) shifted **1 unit left** and **6 units up**.

### Domain and Range:

- **Domain**: Since this is a cubic function, the domain remains \( (-\infty, \infty) \) as the function is defined for all real values of \( x \).
- **Range**: The cubic function \( x^3 \) has a range of \( (-\infty, \infty) \), and since there is only a vertical shift by 6 units, the range of \( g(x) \) also remains \( (-\infty, \infty) \).

### Summary:
- **Horizontal shift**: 1 unit to the left
- **Vertical shift**: 6 units upward
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( (-\infty, \infty) \)

If you would like, I can help you graph the function visually or provide more detailed steps.

Would you like any details or further clarification?

Here are 5 questions to expand your understanding:
1. How does a horizontal shift affect the function's behavior?
2. What happens to the range if a cubic function is vertically stretched?
3. Can the domain of a cubic function ever be restricted by transformations?
4. How would the graph change if the function was \( (x - 1)^3 + 6 \) instead?
5. What role does the coefficient of \( x \) play in stretching or compressing the cubic graph?

**Tip**: Horizontal shifts move the graph left or right, depending on whether the shift is positive or negative.

Starting with the graph of a basic function, graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Find the domain and range of the function. g(x) = (x + 1)^3 + 6

Learn how to graph the cubic function g(x) = (x + 1)^3 + 6 using techniques like shifting and transforming. This solution includes step-by-step instructions on applying horizontal and vertical shifts, and explores the domain and range of the function. Ideal for Grades 10-12 students.

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