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.