Question

the graph shows the height h in feet of a rock dropped down a deep well as a function of time t in seconds. describe what the vertex and endpoint represent, and then determine the equation of the function. the vertex represents the time t = seconds when the rock was released from ground level. the end point represents the height h = feet at which the rock hits the bottom of the well t = seconds after it was released. question 5 of 11

Explanation:

Step1: Identify vertex time

The vertex of the graph is the point \((3.5,- 196)\). In the context of time - height graph for the rock, the \(x\) - coordinate of the vertex represents time. So the time \(t\) when the rock was released from ground level (represented by the vertex) is \(t = 0\) seconds. In a free - fall motion starting from rest, the motion begins at \(t = 0\).

Step2: Analyze endpoint

The endpoint of the graph represents the moment when the rock hits the bottom of the well. From the graph, we can see that the rock hits the bottom of the well at a certain height. Since the graph shows the height as a function of time, the height \(h\) at the endpoint is the height of the bottom of the well. Looking at the graph, as \(t\) increases, the rock reaches the bottom. If we assume the graph is a linear or non - linear function representing the free - fall motion, and we know the general form of the height function for free - fall is \(h(t)=h_0+v_0t-\frac{1}{2}gt^2\) (where \(h_0\) is the initial height, \(v_0\) is the initial velocity and \(g\) is the acceleration due to gravity). Here \(h_0 = 0\) (released from ground level) and \(v_0=0\) (dropped, not thrown). The graph seems to be a quadratic function of the form \(h(t)=- \frac{1}{2}gt^2\). From the graph, when the rock hits the bottom, if we assume the endpoint is at \(t = 4.5\) seconds and \(h=-225\) feet (by observing the grid of the graph).

Answer:

The vertex represents the time \(t = 0\) seconds when the rock was released from ground level.
The endpoint represents the height \(h=-225\) feet at which the rock hits the bottom of the well at \(t = 4.5\) seconds.
The equation of the function for free - fall starting from rest (\(h_0 = 0,v_0 = 0\)) is \(h(t)=-16t^2\) (using \(g = 32\) ft/s², so \(\frac{1}{2}g=16\)). When \(t = 3.5\), \(h=-16\times(3.5)^2=-16\times12.25=-196\) which matches the point \((3.5, - 196)\) on the graph.