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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.

The function f(x) = 1/9 * (x - 7)^2 + 3 gives a metal ball’s height above the ground f(x), in inches, x seconds after it started moving on a track, where 0 ≤ x ≤ 10. Which of the following is the best interpretation of the vertex of the graph of y = f(x) in the xy-plane?

A) The metal ball's minimum height was 3 inches above the ground.
B) The metal ball's minimum height was 7 inches above the ground.
C) The metal ball's height was 3 inches above the ground when it started moving.
D) The metal ball's height was 7 inches above the ground when it started moving.

Explore how to interpret the vertex of a quadratic function using the example of a metal ball’s height. Learn the key concepts of quadratic functions and how to identify the minimum or maximum value. Suitable for Grades 9-11.

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