Question

\begin{aligned}
f(x)&=x^2
\\\\
g(x)&=(x - 6)^2
\end{aligned}

Answer 1

The functions provided are:

\[
f(x) = x^2
\]

\[
g(x) = (x - 6)^2
\]

### Analysis of the Functions

1. **Function \( f(x) \)**:
- This is a standard quadratic function that opens upwards. The vertex is at the origin (0,0).
- The graph of \( f(x) \) is a parabola symmetric about the y-axis.

2. **Function \( g(x) \)**:
- This is also a quadratic function that opens upwards. The vertex is shifted to the right to (6,0).
- The graph of \( g(x) \) is a parabola symmetric about the line \( x=6 \).

### Properties of the Functions

- **Intercepts**:
- For \( f(x) \), the y-intercept is at (0,0).
- For \( g(x) \), the y-intercept occurs when \( x=0 \):
\[
g(0) = (0 - 6)^2 = 36 \quad \Rightarrow \quad (0, 36)
\]

- **Vertex**:
- The vertex of \( f(x) \) is (0,0).
- The vertex of \( g(x) \) is (6,0).

- **Range**:
- The range for both functions is \( [0, \infty) \) since both parabolas open upwards.

- **Transformation**:
- Function \( g(x) \) can be seen as a transformation of \( f(x) \) by shifting \( f(x) \) 6 units to the right.

### Intersection Points

To find the points of intersection, set \( f(x) = g(x) \):

\[
x^2 = (x - 6)^2
\]

Expanding the right side:

\[
x^2 = x^2 - 12x + 36
\]

Subtract \( x^2 \) from both sides:

\[
0 = -12x + 36
\]

Solving for \( x \):

\[
12x = 36 \quad \Rightarrow \quad x = 3
\]

Now substitute \( x = 3 \) back into either function to find the corresponding \( y \)-value:

\[
f(3) = 3^2 = 9
\]

Thus, the point of intersection is:

\[
(3, 9)
\]

### Summary

- The two functions \( f(x) \) and \( g(x) \) are both parabolas opening upwards.
- They intersect at the point \( (3, 9) \).
- The vertex of \( f(x) \) is at (0,0), and the vertex of \( g(x) \) is at (6,0).