91Ó°ÊÓ

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The illumination from a light source varies inversely as the square of the distance from the light source. If you raise a lamp from 15 inches to 30 inches over your desk, what happens to the illumination?

An athlete whose event is the shot put releases the shot with the same initial velocity but at different angles. The figure shows the parabolic paths for shots released at angles of \(35^{\circ}\) and \(65^{\circ} .\) Exercises \(57-58\) are based on the functions that model the parabolic paths. When the shot whose path is shown by the red graph on the previous page is released at an angle of \(65^{\circ},\) its height, \(g(x),\) in feet, can be modeled by $$ g(x)=-0.04 x^{2}+2.1 x+6.1 $$ where \(x\) is the shot's horizontal distance, in feet, from its point of release. Use this model to solve parts (a) through (c) and verify your answers using the red graph. a. What is the maximum height, to the nearest tenth of a foot, of the shot and how far from its point of release does this occur? b. What is the shot's maximum horizontal distance, to the nearest tenth of a foot, or the distance of the throw? c. From what height was the shot released?

A ball is thrown upward and outward from a height of 6 feet. The height of the ball, \(f(x),\) in feet, can be modeled by $$ f(x)=-0.8 x^{2}+3.2 x+6 $$ where \(x\) is the ball's horizontal distance, in feet, from where it was thrown. a. What is the maximum height of the ball and how far from where it was thrown does this occur? b. How far does the ball travel horizontally before hitting the ground? Round to the nearest tenth of a foot. c. Graph the function that models the ball's parabolic path.

In Exercises 41–64, a. Use the Leading Coefficient Test to determine the graph’s end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. $$f(x)=(x-2)^{2}(x+4)(x-1)$$

You have 600 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

The equations in Exercises 72–75 have real roots that are rational. Use the Rational Zero Theorem to list all possible rational roots. Then graph the polynomial function in the given viewing rectangle to determine which possible rational roots are actual roots of the equation. $$ 6 x^{3}-19 x^{2}+16 x-4=0 ;[0,2,1] \text { by }[-3,2,1] $$

Why is a third-degree polynomial function with a negative leading coefficient not appropriate for modeling nonnegative real-world phenomena over a long period of time?

Explain the relationship between the multiplicity of a zero and whether or not the graph crosses or touches the x-axis and turns around at that zero.

If f is a polynomial function, and f(a) and f(b) have opposite signs, what must occur between a and b? If f(a) and f(b) have the same sign, does it necessarily mean that this will not occur? Explain your answer.

Can the graph of a polynomial function have no x-intercepts? Explain.