Question

the graph shows the height of a football after being thrown. use the graph to estimate the x - and y - intercepts of the function, where the function is positive and negative, and interpret the meanings in the context of the situation.
the x - intercept is __. this is __.
the y - intercept is __. this is __.
the function is only __ because __.

Explanation:

Step1: Find the y - intercept

The y - intercept is the value of the function when \(x = 0\). Looking at the graph, when \(x=0\) (time \(t = 0\) seconds), the height \(y\) is \(5\) feet. In the context of the problem, this represents the initial height of the football when it is thrown (at \(t = 0\) seconds).

Step2: Find the x - intercept

The x - intercept is the value of \(x\) when \(y = 0\). From the graph, when \(y = 0\) (height of the football is \(0\) feet), \(x\approx9\) seconds. This represents the time when the football hits the ground.

Step3: Determine the domain of the function

The function is only positive for \(0\lt x\lt9\) because the height of the football (\(y\)) is positive (above the ground) between \(t = 0\) seconds (when it is thrown) and \(t\approx9\) seconds (when it hits the ground).

Answer:

The \(x\) - intercept is \(9\). This is the time when the football hits the ground. The \(y\) - intercept is \(5\). This is the initial height of the football. The function is only positive for \(0\lt x\lt9\) because the football is above the ground (height \(y>0\)) in this time interval.