How to Analyze the Path of a Rocket Using Quadratic Functions

When studying the motion of a rocket launched from the ground, we often model its path using a quadratic expression, such as \(y = -x^2 + 14x - 40\). This equation represents a parabola that opens downward, indicating that the rocket reaches a maximum height before descending. To fully understand the rocket's trajectory, we need to analyze two key features: the vertex of the parabola, which gives us the maximum height, and the x-intercepts, which tell us how far the rocket travels before returning to the ground.

### Step 1: Finding the Maximum Height
The vertex of a parabola described by a quadratic equation \(y = ax^2 + bx + c\) is crucial for understanding its peak performance. The x-coordinate of the vertex can be calculated using the formula: \(x = -\frac{b}{2a}\). In our case, \(a = -1\) and \(b = 14\). Plugging these values into the formula, we get:

\[ x = -\frac{14}{2(-1)} = 7 \]

This means the rocket reaches its maximum height at \(x = 7\) feet. To find the corresponding maximum height, substitute \(x = 7\) back into the original equation:

\[ y = -7^2 + 14(7) - 40 = -49 + 98 - 40 = 9 \]

Thus, the maximum height of the rocket is 9 feet. This result is essential for understanding how high the rocket can go before it starts descending.

### Step 2: Finding the X-Intercepts
Next, we need to determine where the rocket touches the ground, which corresponds to the x-intercepts of the quadratic function. To find these intercepts, we set \(y = 0\) and solve the equation:

\[ 0 = -x^2 + 14x - 40 \]

Rearranging this, we have:

\[ x^2 - 14x + 40 = 0 \]

This quadratic can be factored to:

\[ (x - 10)(x - 4) = 0 \]

From this factored form, we can see that the x-intercepts are \(x = 10\) and \(x = 4\). This means the rocket is on the ground at these points, representing the distance it traveled through the air. The rocket travels a total distance of 10 feet before landing back on the ground.

### Real-World Applications
Understanding the path of a rocket is not only critical in physics but also in various engineering fields, such as aerospace engineering. By mastering these concepts, students can apply quadratic functions to real-world situations, enhancing their problem-solving skills and analytical thinking.

In conclusion, analyzing the path of a rocket using its quadratic model provides valuable insights into its maximum height and flight distance. This knowledge is not just theoretical; it has practical implications in fields that rely on understanding motion and trajectories. By following these steps, students can deepen their comprehension of quadratic functions and their applications in real-life scenarios.