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 4 right parenthesis cubed minus 6

To graph the function \( g(x) = (x + 4)^3 - 6 \), we will begin by identifying the graph of the basic function, which is \( f(x) = x^3 \). Then, we apply the appropriate transformations (shifting, stretching, compressing, and/or reflecting).

### 1. Basic Function: \( f(x) = x^3 \)
The graph of \( f(x) = x^3 \) is a cubic function with:
- An inflection point at the origin \((0, 0)\),
- The curve passing through \((1, 1)\), \((0, 0)\), and \((-1, -1)\),
- The graph increases to the right of the origin and decreases to the left of the origin.

### 2. Apply Transformations to \( g(x) = (x + 4)^3 - 6 \)
We can break this down into two transformations:
- **Horizontal Shift:** The term \( (x + 4) \) means the graph shifts **4 units to the left**.
- **Vertical Shift:** The term \( -6 \) at the end indicates the graph shifts **6 units downward**.

Thus, the graph of \( f(x) = x^3 \) is shifted left by 4 units and then down by 6 units to obtain the graph of \( g(x) = (x + 4)^3 - 6 \).

### 3. Domain and Range
- The **domain** of the function \( g(x) \) is all real numbers because a cubic function is defined for all \( x \):
  \[
  \text{Domain}: (-\infty, \infty)
  \]
- The **range** of the function \( g(x) \) is also all real numbers because the cubic function tends towards \( \infty \) as \( x \to \infty \) and tends towards \( -\infty \) as \( x \to -\infty \):
  \[
  \text{Range}: (-\infty, \infty)
  \]

### Summary of Transformations:
- The graph of \( f(x) = x^3 \) shifts **4 units left** and **6 units down** to form the graph of \( g(x) = (x + 4)^3 - 6 \).

Would you like to see the graph, or do you have any questions? 

Here are 5 related questions to expand on the topic:
1. How do horizontal and vertical reflections affect the graph of a cubic function?
2. How would compressing or stretching the function vertically affect its shape?
3. What happens to the domain and range if we add an absolute value to \( g(x) \)?
4. How would adding or subtracting different constants inside the cubic term affect the transformation?
5. How would the graph change if \( g(x) = - (x + 4)^3 - 6 \)?

**Tip:** Always apply horizontal shifts first before vertical shifts when graphing transformations.

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 + 4)^3 - 6

Learn how to graph the function g(x) = (x + 4)^3 - 6 by applying transformations such as horizontal and vertical shifts. This problem covers key concepts in algebra, including cubic functions, and explores the domain and range of the function. Suitable for students in Grades 9-11.

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