Analyzing model in vertex form

TL;DR

An object is launched from a platform and its height is modeled using a quadratic equation in vertex form.

Transcript

• [Sal] An object is launched from a platform. Its height in meters, x seconds after the launch, is modeled by: h of x is equal to negative five times x minus four squared plus 180. Normally, when they talk about seconds or time, they usually would use the variable t, but we can roll with x being that. Let's think about what's going to happen here.... Read More

Key Insights

• 🚀 The equation h(x) = -5x^2 + 180 models the height of an object launched from a platform over time.
• 💁 The vertex form of the equation reveals the shape of the trajectory as a downward-opening parabola.
• 🤒 Evaluating h(0) gives the height of the platform, which is 100 meters in this case.
• ☺️ The maximum height is achieved when x = 4 seconds, and the height at that time is 180 meters.

Questions & Answers

Q: How do we know that the object's trajectory will form a downward-opening parabola?

The equation is in vertex form with a negative coefficient on the x^2 term, indicating a downward opening. Multiplying out the terms confirms this shape.

Q: What is the height of the platform?

By evaluating h(0), we substitute x with 0 in the equation h(x) = -5x^2 + 180, resulting in 100 meters as the height of the platform.

Q: When does the object reach its maximum height?

The maximum height corresponds to the vertex of the parabola. In this case, the x-coordinate of the vertex is 4 seconds, indicating that the maximum height is reached 4 seconds after launch.

Q: How long after launch does the object hit the ground?

Setting h(x) = 0 and solving the equation -5x^2 + 180 = 0, we find that the object hits the ground at x = 10 seconds.

Summary & Key Takeaways

• The object's height is modeled by the equation h(x) = -5x^2 + 180, where x represents time in seconds.

• The platform's height is determined to be 100 meters by evaluating h(0).

• The maximum height is achieved when x = 4 seconds, and the height at that time is 180 meters.

• The object hits the ground 10 seconds after launch.