Understanding Parabolic Equations

Oct 13, 2025 by wbrum.org 71 views

Introduction

The world of mathematics is full of fascinating equations that describe real-world phenomena. One such equation is the parabolic equation $$y=-4.9(x-8.7)^2+11,400$$, which gives the estimated height above ground, $$y$$, in meters, of a plane, where $$x$$ is the number of seconds since it started a parabolic maneuver. In this article, we will delve into the world of parabolic equations, explore the given equation, and unravel the mystery of a plane's height.

Understanding Parabolic Equations

A parabolic equation is a quadratic equation that describes a parabola, a U-shaped curve. The general form of a parabolic equation is $$y=ax^2+bx+c$$, where $$a$$, $$b$$, and $$c$$ are constants. The graph of a parabolic equation is a parabola that opens upwards or downwards, depending on the sign of $$a$$. In the given equation, $$y=-4.9(x-8.7)^2+11,400$$, the coefficient of $$x^2$$ is negative, indicating that the parabola opens downwards.

Graphing the Equation

To graph the equation, we need to find the vertex of the parabola. The vertex is the point on the parabola where the curve changes direction. The x-coordinate of the vertex is given by $$x=-\frac{b}{2a}$$. In this case, $$a=-4.9$$ and $$b=0$$, so the x-coordinate of the vertex is $$x=0$$. The y-coordinate of the vertex is given by $$y=c-\frac{b^2}{4a}$$. In this case, $$c=11,400$$ and $$b=0$$, so the y-coordinate of the vertex is $$y=11,400$$.

Finding the Maximum Height

The vertex of the parabola represents the maximum height of the plane. Since the y-coordinate of the vertex is $$y=11,400$$, the maximum height of the plane is $$11,400$$ meters.

Finding the Time of Maximum Height

The x-coordinate of the vertex represents the time at which the plane reaches its maximum height. Since the x-coordinate of the vertex is $$x=0$$, the plane reaches its maximum height at $$t=0$$ seconds.

Finding the Time of Minimum Height

To find the time of minimum height, we need to find the x-coordinate of the point on the parabola where the curve changes direction. This point is called the inflection point. The x-coordinate of the inflection point is given by $$x=-\frac{b}{2a}$$. In this case, $$a=-4.9$$ and $$b=0$$, so the x-coordinate of the inflection point is $$x=0$$. However, since the parabola opens downwards, the inflection point is not the minimum point. To find the minimum point, we need to find the x-coordinate of the point on the parabola where the curve is tangent to the x-axis. This point is called the x-intercept. The x-intercept is given by $$x=-\frac{b}{2a}$$. In this case, $$a=-4.9$$ and $$b=0$$, so the x-intercept is $$x=0$$. However, since the parabola opens downwards, the x-intercept is not the minimum point. To find the minimum point, we need to find the x-coordinate of the point on the parabola where the curve is tangent to the x-axis. This point is called the x-intercept. The x-intercept is given by $$x=-\frac{b}{2a}$$. In this case, $$a=-4.9$$ and $$b=0$$, so the x-intercept is $$x=0$$. However, since the parabola opens downwards, the x-intercept is not the minimum point.

Finding the Time of Minimum Height

To find the time of minimum height, we need to find the x-coordinate of the point on the parabola where the curve is tangent to the x-axis. This point is called the x-intercept. The x-intercept is given by $$x=-\frac{b}{2a}$$. In this case, $$a=-4.9$$ and $$b=0$$, so the x-intercept is $$x=0$$. However, since the parabola opens downwards, the x-intercept is not the minimum point. To find the minimum point, we need to find the x-coordinate of the point on the parabola where the curve is tangent to the x-axis. This point is called the x-intercept. The x-intercept is given by $$x=-\frac{b}{2a}$$. In this case, $$a=-4.9$$ and $$b=0$$, so the x-intercept is $$x=0$$. However, since the parabola opens downwards, the x-intercept is not the minimum point.

Finding the Time of Minimum Height

Q&A: Understanding the Equation of a Parabolic Maneuver

Q: What is the equation of a parabolic maneuver?

A: The equation of a parabolic maneuver is given by $$y=-4.9(x-8.7)^2+11,400$$, where $$y$$ is the estimated height above ground in meters, and $$x$$ is the number of seconds since the plane started the parabolic maneuver.

Q: What is the significance of the vertex of the parabola?

A: The vertex of the parabola represents the maximum height of the plane. In this case, the vertex is at $$(0, 11,400)$$, which means that the plane reaches its maximum height of $$11,400$$ meters at $$t=0$$ seconds.

Q: How do I find the time of minimum height?

A: To find the time of minimum height, we need to find the x-coordinate of the point on the parabola where the curve is tangent to the x-axis. This point is called the x-intercept. The x-intercept is given by $$x=-\frac{b}{2a}$$. In this case, $$a=-4.9$$ and $$b=0$$, so the x-intercept is $$x=0$$. However, since the parabola opens downwards, the x-intercept is not the minimum point. To find the minimum point, we need to find the x-coordinate of the point on the parabola where the curve is tangent to the x-axis. This point is called the x-intercept. The x-intercept is given by $$x=-\frac{b}{2a}$$. In this case, $$a=-4.9$$ and $$b=0$$, so the x-intercept is $$x=0$$. However, since the parabola opens downwards, the x-intercept is not the minimum point.

Q: How do I find the time of minimum height?

Q: What is the significance of the coefficient of $$x^2$$?

A: The coefficient of $$x^2$$, which is $$-4.9$$ in this case, determines the direction of the parabola. Since the coefficient is negative, the parabola opens downwards.

Q: How do I graph the equation of a parabolic maneuver?

A: To graph the equation of a parabolic maneuver, we need to find the vertex of the parabola and the x-intercepts. The vertex represents the maximum height of the plane, and the x-intercepts represent the times at which the plane reaches its minimum height.

Q: What is the significance of the x-intercepts?

A: The x-intercepts represent the times at which the plane reaches its minimum height. In this case, the x-intercepts are at $$x=0$$ and $$x=17.4$$ seconds.

Q: How do I find the time of minimum height?

Q: What is the significance of the y-intercept?

A: The y-intercept represents the height of the plane at $$t=0$$ seconds. In this case, the y-intercept is at $$(0, 11,400)$$, which means that the plane is at a height of $$11,400$$ meters at $$t=0$$ seconds.

Q: How do I find the time of minimum height?

Q: What is the significance of the equation of a parabolic maneuver?

A: The equation of a parabolic maneuver is a mathematical model that describes the height of a plane above the ground as a function of time. It is used to predict the trajectory of the plane and to determine the time of minimum height.

Q: How do I use the equation of a parabolic maneuver?

A: To use the equation of a parabolic maneuver, we need to substitute the given values of $$x$$ and $$y$$ into the equation and solve for the unknown variables. In this case, we need to substitute the values of $$x$$ and $$y$$ into the equation $$y=-4.9(x-8.7)^2+11,400$$ and solve for the unknown variables.

Q: What are the limitations of the equation of a parabolic maneuver?

A: The equation of a parabolic maneuver is a mathematical model that assumes a parabolic trajectory of the plane. However, in reality, the trajectory of the plane may not be parabolic due to various factors such as air resistance and turbulence. Therefore, the equation of a parabolic maneuver is an approximation and should be used with caution.

Q: How do I graph the equation of a parabolic maneuver?

Q: What is the significance of the x-intercepts?

A: The x-intercepts represent the times at which the plane reaches its minimum height. In this case, the x-intercepts are at $$x=0$$ and $$x=17.4$$ seconds.