22 POINTS The height of a pop fly ball over the time after the ball is hit is modeled by the function, h(t) = -16t2 + 81t + 5, where h is the ball's height in feet above the ground and t is the time in seconds after the ball is hit.

How far above the ground was the ball when it was initially hit?

5 feet

Step-by-step explanation:

In the height equation (notice the ball's horizontal distance traveled  is not addressed here), each term means something.

'5' is where the ball started from, since that makes a difference where it winds up t seconds later. If you have trouble visualizing that, imagine if the ball was batted from the top of a skyscraper. It would take a lot longer to hit the ground than if you hit it from only 5 feet high off the ground!

81t means the ball was hit with enough force to give it an initial upward velocity of 81 feet per unit of t, as it travels forward, so that's 81 feet per second. (Depending on the angle the ball was struck, the horizontal component may be a different number.)

-16t² accounts for Earth gravity pulling the ball downward the whole time, t, the ball is in the air (the negative sign is for downward)

For 22 points, I figured you deserved a very thorough explanation.

Hope this helps!

the range of a function is the range of y-values for which it is defined, or exists. given that we are only looking at the y-values, we can already discount the first two answers as they deal with the domain of the function (range of x-values).

now, what we need to do is find the smallest and highest y-values for which the function is defined. if you'd like, you could take a ruler and start at the very bottom of the graph, find the lowest point, then move the ruler higher and higher until you reach the highest point. doing so either in your head or with an actual ruler, we can see that the lowest y-value is -5 and the highest y-value is -1.

the question now remains as to whether they are included in the range (fourth answer) or excluded (third answer). the key here is to recognise whether they are marked with an open or a closed circle. since both (-2, -5) and (2, -1) are marked with a closed circle, they are points that exist on the graph and are thus included in the range (note that if they were marked with an open circle like (4, -2) then that would signify that there is a break in the graph at that point).

thus, the fourth answer, (y | -5 ≤ y ≤ -1), is correct.

also note that usually it is good practice to scan your eye inbetween the highest and lowest point for any breaks in the graph - we do have one here at (4, -2), however -2 is still included in the range because there is another point that is included in the graph with the same y-value, ie. point (1, -2).

i hope that but if you require any further explanation don't hesitate to comment below.