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 x squared plus 5
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Part 1
Use the graphing tool to graph the equation.



Let's break down the transformation of the basic function \( f(x) = x^2 \) into \( g(x) = x^2 + 5 \).

### Steps to Graph the Function:

1. **Start with the basic graph**: The graph of \( f(x) = x^2 \) is a parabola that opens upwards with its vertex at the origin (0, 0).

2. **Vertical Shift**: The term \( +5 \) in \( g(x) = x^2 + 5 \) indicates a **vertical shift**. Specifically, the entire graph of \( f(x) \) is shifted **upward by 5 units**. This means that the vertex of the parabola moves from (0, 0) to (0, 5).

3. **No other transformations**: There is no horizontal shift, reflection, or stretching/compression involved.

### Domain and Range:

- **Domain**: The domain of \( g(x) = x^2 + 5 \) is the set of all real numbers, since \( x^2 \) is defined for all real \( x \). 
  \[
  \text{Domain: } (-\infty, \infty)
  \]

- **Range**: Since the parabola opens upwards and the minimum value of \( g(x) \) is 5 (at \( x = 0 \)), the range starts at 5 and extends to infinity.
  \[
  \text{Range: } [5, \infty)
  \]

### Summary of the Transformation:

- Start with \( f(x) = x^2 \) (a parabola).
- Shift the graph 5 units upwards to get \( g(x) = x^2 + 5 \).

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If you'd like, I can also generate a visual graph of the function. Would you like that?

### Relative Questions:
1. What would happen if the equation was \( g(x) = x^2 - 5 \)?
2. How would the graph change if the function were \( g(x) = (x - 2)^2 + 5 \)?
3. What would be the domain and range of \( g(x) = -x^2 + 5 \)?
4. What effect does multiplying \( x^2 \) by a constant have on the graph?
5. How do horizontal shifts affect the domain and range of a quadratic function?

### Tip:
Remember that vertical shifts affect the **range** of the function, while horizontal shifts do not.

Learn how to graph the quadratic function g(x) = x^2 + 5 using shifting, compressing, stretching, and reflecting techniques. Understand the domain and range of the function with detailed explanations and examples. Suitable for high school students studying functions and graphing.

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