Fred throws a stone from a bridge into a river below. The stone's height (in meters) above the water, x seconds after Fred threw it, is modeled by the quadratic function H(x) = -5x² + 10x + 15. This equation describes a parabola, a common shape for objects under the influence of gravity. The negative coefficient...
Graphing the Quadratic Function
To visualize the stone's trajectory, we can graph the function H(x) = -5x² + 10x + 15. The graph will show the height of the stone at different times. The x-axis represents time (in seconds) and the y-axis represents the height (in meters). A graphing calculator or software can be used to create this graph easily. The resulting parabola will clearly show the maximum height reached by the stone and the time it takes to hit the water.
[A description of how to graph the function manually or using software would go here, including key points like the vertex, x-intercepts, and y-intercept. Include a visual representation of the graph – perhaps a placeholder image tag or a description explaining what the graph would look like. For example: "The parabola opens downwards, reaching a maximum height at its vertex. The x-intercepts represent the time when the stone hits the water, and the y-intercept represents the initial height of the stone."]
Interpreting the y-intercept
The y-intercept is the point where the graph intersects the y-axis, meaning it represents the height of the stone when x = 0 (at the moment Fred throws it). In this equation, when x=0, H(0) = -5(0)² + 10(0) + 15 = 15 meters. Therefore, the y-intercept (0, 15) indicates that Fred threw the stone from a height of 15 meters above the water.
Interpreting the x-intercepts
The x-intercepts are the points where the graph intersects the x-axis, representing the times when the height of the stone is 0 (when it hits the water). To find the x-intercepts, we set H(x) = 0 and solve the quadratic equation: -5x² + 10x + 15 = 0. This can be solved using the quadratic formula or factoring. [Show the steps to solve the quadratic equation, perhaps using the quadratic formula and arriving at two solutions for x. One will be a negative value (unrealistic in this context) and the other a positive value representing the time the stone hits the water]. The positive x-intercept represents the time it takes for the stone to hit the water. The negative x-intercept is not physically meaningful in this context, as time cannot be negative.
Interpreting the Vertex
The vertex of the parabola is the highest point on the graph, representing the maximum height the stone reaches and the time at which it reaches this height. The x-coordinate of the vertex represents the time at which the stone reaches its maximum height. This can be found using the formula x = -b / 2a for a quadratic equation in the form ax² + bx + c. [Show the calculation using the provided values a=-5 and b=10. Then substitute this x value back into the original equation H(x) to find the y-coordinate of the vertex, which represents the maximum height]. The vertex coordinates (x, y) give both the time (x) at maximum height and the maximum height (y) itself.
Real-world Applications and Limitations of the Model
This model provides a simplified representation of the stone's trajectory. Several factors are not considered, such as air resistance, which would cause the stone to fall slightly slower than predicted. Also, the model assumes the stone's initial velocity is purely vertical; in reality, Fred likely imparted some horizontal velocity as well. Despite these limitations, the quadratic model provides a good approximation of the stone's motion and allows for insightful analysis of key aspects of the trajectory. This same model finds application in a variety of other scenarios involving projectile motion, including the launch of rockets or the trajectory of a ball thrown in sports.
Further Exploration: Variations and Extensions
We can extend this analysis by considering different scenarios. For instance, what if Fred throws the stone with a different initial velocity? How would this affect the graph and the values of the y-intercept, x-intercepts, and vertex? Further, exploring how air resistance might be incorporated into a more complex model can show the limitations of simpler models. [Here you could briefly mention more complex models involving calculus to show the progression of understanding the system]. The study of projectile motion offers countless avenues for further investigation.