1. If a ball falls from a cliff 144 feet high, then its height h after t seconds will be given by $ h(t) = 144 - 16t^{2} $ . This function’s graph is shown at right.

  1. How many seconds will pass before the ball hits the ground?

  2. How high will the ball be after 1 second? After 2 seconds?

  3. When will the ball be exactly 100 feet high? Give an exact answer and an approximation to the nearest hundredth of a second.

  4. When will the ball be exactly 10 feet high? Give an exact answer and an approximation to the nearest hundredth of a second.

  5. As the ball begins its fall, a remarkably nimble insect begins scaling the cliff at a constant speed of 6 feet per second. How high will the insect be when the ball passes it? (Give your answer to the nearest inch.)

  1. The figure at right shows the graph of a function f, and a line through two points on the graph.

  1. Find the coordinates of P and Q (in terms of x and h).

  2. Use your answers from part (a) to write an expression for the slope of line PQ. This should look familiar. (If it doesn’t, try multiplying the top and bottom by -1.)

  1. The equation of the unit circle, as we’ve seen earlier, is $ x^{2} + y^{2} = 1 $ .
  1. The following reasoning may initially look correct, but it is subtly flawed. Identify the flaw:

If we solve for y in the unit circle’s equation, we get $ y = $ . Hence, the graph of $ f(x) = $ must be the unit circle.

  1. What actually is the graph of $ (x) = $ ?

  2. Let $ (x,0) $ be a variable point on the x-axis. Let PQ be the chord of the circle that passes through this variable point and is perpendicular to the x-axis. (See the figure at right.) Express this chord’s length as a function of x.

  3. Let N be the fixed point $ (0,1) $ , and think of P as a variable point that can slide around on the top half of the unit circle. If you’ve studied enough geometry to do this, prove that NP’s perpendicular bisector must pass through the origin.

[Hint: Start with the line through the origin and NP’s midpoint, and prove that it is NP’s perpendicular bisector.]

  1. Express the slope of NP’s perpendicular bisector as a function of x. Then find this function’s domain and range.
  1. Given an equation involving x and y, we can sometimes “disentangle” the two variables to express y as a function of x. (For example, we can rewrite the equation xy = 7 as y = 7/x.) But often we can’t do this. As a particular example, I claim that it is impossible to rewrite the equation

\[ x^{3}+4x^{2}y+9xy^{2}-36x-9y^{3}+36y=0 \]

in an equivalent form in which y is expressed as a function of x. Not merely difficult, mind you, but impossible. Your problem: Explain why this is so.

Hint: You can’t graph this equation by hand, but a computer can do it for you. The insight it gives you will help you solve this problem. (And remember, equivalent equations have the same graph.)