1. Ball Trajectory

A child throws a ball from the window of a house. The height of the ball, h(x), depends on the horizontal distance, x, from the window. Units are in feet. The function in expanded form is given by

h(x) = −4x2 + 8x + 5

a) Does this function have a mathematical domain restriction? Why or why not?

b) Factor the right side of the function (remember the GCF!).

c) Find the following function values: h(0), h(1), h(− 1/ 2 ) , h( 5/ 2 ). Do not use a calculator. Substitute into either the expanded or factored form, whichever is easier.

d) Using DESMOS, graph the function and identify the coordinate points that correspond to the function values in part c). Which of these 4 coordinate points are in the practical domain of the function? Interpret their meaning.

e) State the practical domain and range of this function in interval notation.

f) On the practical domain, state the interval for x in which the height of the ball is decreasing.

g) Refer to your DESMOS graph and state the interval for x in which the ball is 7 feet or more above the ground. To do this, add the line y = 7 to the function list and find the point of intersection to the nearest tenth. 

2. Murrelet Population Study

The Ancient Murrelet is a plucky little bird of the auk family that lives in colonies on islands off the northern Pacific coasts of Russia, Alaska and Canada. Since these birds lay their eggs in burrows on the forest floor, the nestlings are highly susceptible to predation. Their populations have been greatly reduced over the past century by introduced mammalian predators (deer, raccoons, rats). Data has been gathered over several years in order to develop a conservation plan. The following function describes the observable population decline:

where P(t) is the island population in thousands and t is the time in years since January 1, 2010, when the study was started. 

a) Does this function have a mathematical domain restriction? Why or why not?

b) Using DESMOS, graph the function and identify the coordinate points that correspond to the following function values: P(0), P(5), P(−4.3), P(−5) (to the nearest tenth). Which of these 4 coordinate points are in the practical domain of the function? Interpret their meaning.

c) Refer to your DESMOS graph and find the coordinates of the t-intercept. Interpret the meaning of the t-intercept in the problem context.

d) State the practical domain and range of this function in interval notation.

e) Refer to your DESMOS graph and find the year when the population is reduced to 1000 birds. To do this, add the line y = 1 to the function list and find the point of intersection to the nearest tenth.