Starting with the graph of a basic​ function, graph the following function using the techniques of​ shifting, compressing,​ stretching, and/or reflecting. Be sure to identify at least three key points. Find the domain and range of the function. 

g left parenthesis x right parenthesis equals left parenthesis x plus 4 right parenthesis cubed plus 6g(x)=(x+4)3+6

To graph the function \( g(x) = (x + 4)^3 + 6 \), let's follow the process of shifting, stretching, compressing, and reflecting starting with the graph of the basic cubic function \( f(x) = x^3 \).

### Step-by-Step Graphing Process:
1. **Start with the base function**: The base function is \( f(x) = x^3 \), which is a standard cubic curve that passes through the origin \((0, 0)\) with the following key points:
   - \( (-1, -1) \)
   - \( (0, 0) \)
   - \( (1, 1) \)
   
   It has an "S" shape, with increasing behavior for \( x > 0 \) and decreasing for \( x < 0 \).

2. **Shift left by 4 units**: The term \( (x + 4) \) means we shift the graph of \( f(x) = x^3 \) 4 units to the left. So, the key points are now:
   - \( (-5, -1) \) (formerly \( (-1, -1) \))
   - \( (-4, 0) \) (formerly \( (0, 0) \))
   - \( (-3, 1) \) (formerly \( (1, 1) \))
   
3. **Shift up by 6 units**: The \( +6 \) at the end means we shift the entire graph up by 6 units. The key points after this shift are:
   - \( (-5, -1 + 6) = (-5, 5) \)
   - \( (-4, 0 + 6) = (-4, 6) \)
   - \( (-3, 1 + 6) = (-3, 7) \)

### Domain and Range:
- **Domain**: The domain of \( g(x) \) is the set of all real numbers since a cubic function is defined for all \( x \in \mathbb{R} \). Hence, the domain is \( (-\infty, \infty) \).
  
- **Range**: The range of \( g(x) \) is also all real numbers, as the cubic function takes on all values from negative infinity to positive infinity. Thus, the range is \( (-\infty, \infty) \).

### Key Points of the Transformed Function:
- \( (-5, 5) \)
- \( (-4, 6) \)
- \( (-3, 7) \)

Would you like more details or have any questions? 

Here are five related questions for further practice:
1. How would the graph change if the function was \( g(x) = -(x + 4)^3 + 6 \)?
2. What if the function was \( g(x) = (x + 4)^3 - 6 \)? How would the range be affected?
3. What are the inflection points of the basic cubic function \( f(x) = x^3 \)?
4. How do you find the derivative of \( g(x) = (x + 4)^3 + 6 \)?
5. Can you generalize the transformation for a function \( g(x) = (x + a)^3 + b \)?

**Tip**: Always shift horizontally first before performing any vertical transformations when dealing with function transformations.

Starting with the graph of a basic function, graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Be sure to identify at least three key points. Find the domain and range of the function.
g(x) = (x + 4)^3 + 6

Learn how to graph the cubic function g(x) = (x + 4)^3 + 6 by using techniques of shifting, including a step-by-step process with key points identified. Explore the domain and range of the function. Suitable for students in Grades 10-12.

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