How is the graph of $$ f(x)=(x-3)^{2}+7 $$ generated from the graph of $$ g(x)=x^{2} $$ ? A. Translate $$ g(x) 3 $$ units right and 7 units downward to generate $$ f(x) $$ B. Translate $$ g(x) 3 $$ units right and 7 units upward to generate $$ f(x) $$ C. Translate $$ g(x) 3 $$ units left and 7 units upward to generate $$ f(x) $$ D. Translate $$ g(x) 3 $$ units left and 7 units downward to generate $$ f(x) $$

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UpStudy AI · Accepted Answer

To determine how the graph of $$ f(x) = (x - 3)^2 + 7 $$ is generated from the graph of $$ g(x) = x^2 $$, let's analyze the transformations step by step.

### Understanding the Transformation

The general form for transforming a function $$ g(x) $$ is:

$$
f(x) = g(x - h) + k
$$

Here:
- **$$ h $$** represents a horizontal shift.
  - If $$ h $$ is positive, the graph shifts **$$ h $$ units to the right**.
  - If $$ h $$ is negative, the graph shifts **$$ |h| $$ units to the left**.
- **$$ k $$** represents a vertical shift.
  - If $$ k $$ is positive, the graph shifts **$$ k $$ units upward**.
  - If $$ k $$ is negative, the graph shifts **$$ |k| $$ units downward**.

### Applying to the Given Functions

For $$ f(x) = (x - 3)^2 + 7 $$:

- Compare this with the general form:
  - $$ h = 3 $$: This means the graph of $$ g(x) = x^2 $$ shifts **3 units to the right**.
  - $$ k = 7 $$: This means the graph shifts **7 units upward**.

### Conclusion

The graph of $$ f(x) $$ is obtained by first shifting the graph of $$ g(x) = x^2 $$ **3 units to the right** and then **7 units upward**.

**Answer:**  
**B. Translate $$ g(x) 3 $$ units right and 7 units upward to generate $$ f(x) $$**
The graph of $$ f(x) = (x - 3)^2 + 7 $$ is generated by translating the graph of $$ g(x) = x^2 $$ **3 units to the right** and **7 units upward**. **Answer:** B. Translate $$ g(x) 3 $$ units right and 7 units upward to generate $$ f(x) $$

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