The given function is $f(x) = \frac{1}{9}(x - 7)^2 + 3$. This function represents the height of a metal ball above the ground in inches, $x$ seconds after it started moving on a track.

This function is in the vertex form of a quadratic equation, which is $f(x) = a(x - h)^2 + k$. In this form:

• $(h, k)$ represents the vertex of the parabola.
• $a$ indicates the direction of the parabola (upward if $a > 0$ and downward if $a < 0$).

For the given function:

• $h = 7$
• $k = 3$
• $a = \frac{1}{9}$ (which is positive, so the parabola opens upwards)

The vertex of the function is at the point $(7, 3)$, meaning that when $x = 7$ seconds, the height of the metal ball is 3 inches. Since the parabola opens upwards, this point represents the minimum height of the ball.

Therefore, the correct interpretation is:

A) The metal ball's minimum height was 3 inches above the ground.

Would you like any more details or have any questions?

Here are 5 related questions to explore further:

1. How does changing the value of $h$ in the vertex form $f(x) = \frac{1}{9}(x-h)^2 + k$ affect the graph?
2. What happens to the graph if the coefficient $a$ in $f(x) = a(x - h)^2 + k$ becomes negative?
3. How would the vertex and the minimum/maximum height change if the value of $k$ in the function is altered?
4. How does the range of the function change based on different values of $a$, $h$, and $k$?
5. What would be the interpretation of the vertex if the function represented the height of the ball over time with $a < 0$?

Tip: In quadratic functions, the vertex can be quickly identified in the vertex form $f(x) = a(x-h)^2 + k$, where $(h, k)$ directly gives the vertex coordinates.