When the quadratic functions below are graphed in the $$ x y $$-plane, is the vertex for each function above the $$ x $$-axis, below the $$ x $$-axis, or on the $$ x $$-axis? I. $$ f(x)=x^{2}-5^{2} $$ II. $$ f(x)=x^{2}+5^{2} $$ III. $$ f(x)=(x-5)^{2} $$

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To determine the position of the vertex for each quadratic function relative to the $$ x $$-axis, we can analyze each function individually.

### I. $$ f(x) = x^2 - 5^2 $$

This can be rewritten as:
$$
f(x) = x^2 - 25
$$
The vertex of a quadratic function in the form $$ f(x) = ax^2 + bx + c $$ is given by the point $$ (h, k) $$, where $$ h = -\frac{b}{2a} $$ and $$ k = f(h) $$. In this case, $$ a = 1 $$, $$ b = 0 $$, and $$ c = -25 $$.

The vertex occurs at:
$$
h = -\frac{0}{2 \cdot 1} = 0
$$
Now, we find $$ k $$:
$$
k = f(0) = 0^2 - 25 = -25
$$
Thus, the vertex is at $$ (0, -25) $$, which is **below the $$ x $$-axis**.

### II. $$ f(x) = x^2 + 5^2 $$

This can be rewritten as:
$$
f(x) = x^2 + 25
$$
Here, $$ a = 1 $$, $$ b = 0 $$, and $$ c = 25 $$.

The vertex occurs at:
$$
h = -\frac{0}{2 \cdot 1} = 0
$$
Now, we find $$ k $$:
$$
k = f(0) = 0^2 + 25 = 25
$$
Thus, the vertex is at $$ (0, 25) $$, which is **above the $$ x $$-axis**.

### III. $$ f(x) = (x - 5)^2 $$

This function is already in vertex form, where the vertex is given directly by the expression $$ (h, k) $$. Here, $$ h = 5 $$ and $$ k = 0 $$.

Thus, the vertex is at $$ (5, 0) $$, which is **on the $$ x $$-axis**.

### Summary of Vertex Positions:
- I. Vertex is below the $$ x $$-axis.
- II. Vertex is above the $$ x $$-axis.
- III. Vertex is on the $$ x $$-axis.
I. Vertex is below the $$ x $$-axis. II. Vertex is above the $$ x $$-axis. III. Vertex is on the $$ x $$-axis.

When the quadratic functions below are graphed in the \( x y \)-plane, is the vertex for each function above the \( x \)-axis, below the \( x \)-axis, or on the \( x \)-axis? I. \( f(x)=x^{2}-5^{2} \) II. \( f(x)=x^{2}+5^{2} \) III. \( f(x)=(x-5)^{2} \)

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