This problem involves finding the equation of a parabola in graphing form (y = a(x - h)^2 + k) that models the trajectory of a football kicked by an Eagles kicker. We're given that the football needs to travel 60 yards horizontally and reach a maximum height of 36 yards. Our goal is to determine the value of 'a' in the...
Key Concepts
Parabolas and Their Equations
A parabola is a symmetrical curve shaped like a U or an upside-down U. Its equation in graphing form is:
y = a(x - h)^2 + k
where:
- (h, k) represents the vertex of the parabola (the point where the parabola changes direction).
- 'a' determines the shape of the parabola:
- If 'a' is positive, the parabola opens upwards.
- If 'a' is negative, the parabola opens downwards.
- The larger the absolute value of 'a', the narrower the parabola; the smaller the absolute value of 'a', the wider the parabola.
Finding the Vertex
In our problem, the maximum height of the football corresponds to the vertex of the parabola. Since the football travels 60 yards horizontally, the vertex is located at the midpoint of this distance, which is 30 yards. Therefore, the vertex is (30, 36).
Solving the Problem
1. **Substitute the vertex into the equation:**
y = a(x - 30)^2 + 36
2. **Use the given information about the football's path to find 'a':**
We know the football lands 60 yards away, meaning when x = 60, y = 0. Substitute these values into the equation:
0 = a(60 - 30)^2 + 36
3. **Solve for 'a':**
0 = a(30)^2 + 36
0 = 900a + 36
-36 = 900a
a = -36/900
a = -1/25
Conclusion
Therefore, the 'a' value of the parabolic arc of the football is -1/25. This negative value indicates that the parabola opens downwards, which is consistent with the trajectory of a football kicked into the air.
Further Exploration
The equation of the parabola representing the football's trajectory is now complete:
y = (-1/25)(x - 30)^2 + 36
Using this equation, we can calculate the height of the football at any horizontal distance. This information could be valuable for analyzing the kicker's technique and optimizing the kick for maximum distance and accuracy.