Question

1. an object is thrown up in the air from the top of a building. the relationship between time and height can be modelled by a quadratic equation. what would each of the following represent? (4 marks) a) the x-coordinate of the vertex

b) the y-coordinate of the vertex
c) the x-intercepts
d) the y-intercept

Explanation:

Part (a)

In the quadratic model for the object's height (\(y\)) over time (\(x\)), the vertex of the parabola (which opens downward here) represents the maximum point. The \(x\)-coordinate of the vertex gives the time at which the object reaches its maximum height. This is because the \(x\)-axis represents time, so the \(x\)-value of the vertex corresponds to the specific time when the height is maximized.

Since the quadratic models height (\(y\)) over time (\(x\)), and the vertex is the maximum point of the parabola (as the object is thrown upward and then falls), the \(y\)-coordinate of the vertex corresponds to the maximum height the object attains. The \(y\)-axis represents height, so this value is the peak height of the object's trajectory.

The \(x\)-intercepts of a graph are the points where \(y = 0\) (height = 0). In the context of the object's motion, when \(y = 0\), the object is at ground level. So, the \(x\)-intercepts represent the times when the object is at a height of 0 (ground level). One intercept will be at time \(t = 0\) (when it's thrown from the building, but wait, no—actually, the building has a height, so the \(y\)-intercept is the initial height. Wait, correction: the object is thrown from the top of a building, so the initial height is the \(y\)-intercept. The \(x\)-intercepts: one is when it's thrown (but no, at \(t = 0\), it's at the building's height, so the other \(x\)-intercept is when it hits the ground after being thrown up and coming down, and maybe a negative \(x\)-intercept which doesn't make sense in time, so we consider the positive \(x\)-intercept as the time when it lands on the ground. So generally, the \(x\)-intercepts (specifically the positive one) represent the time(s) when the object is at ground level (height = 0).

Answer:

The \(x\)-coordinate of the vertex represents the time at which the object reaches its maximum height.

Part (b)