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Appreciate doable workings if possible (trying to help a kid with Math I have not done for a long long time):

1.  Given:  Projectile launched straight into the air from 0 level (ground).  V (initial) = 256ft/s.  When will it be 768 ft from ground?  Isn't it just t=d/r? Or is there more to it?  *Now discussing final answer

2.  Identify intervals on which the function is increasing, decreasing, or constant:

a.  f(x) = |x+2| -1
b.  g(x) = x^3 - x^2 -2x

.....more to come....

Thanks much!

I know but I was thinking it was too simple a solution for a pre-calc class, tho the teacher might just be tricking them since it was the last question.

If there is gravity...

    x = x₀ + v₀t + (1/2)gt²

Plugging in numbers (assuming fired from the surface of teh Earth)...

    768 = 0 + 256 * t - 1/2 32 t²  => t² - 16t + 48 = 0

Using the quadratic equation we get the answers of 4 and 12 seconds.

So Matt, considering it's a Prec-calc class should I assume that the teacher has asked a question that requires your solution?

I am guessing so. This is a fairly straight-forward quadratic equation. I regularly used similar things both as homework and test questions in algebra classes I taught. (The fact that the answers come out to be nice round numbers also would lead me to guess that this is what was being asked for.)

The first one would have 2 answers because it will be at the height twice (once on the way up and again when gravity pulls it back down).

If there is gravity...

    x = x₀ + v₀t + (1/2)gt²

Plugging in numbers (assuming fired from the surface of teh Earth)...

    768 = 0 + 256 * t - 1/2 32 t²  => t² - 16t + 48 = 0

Using the quadratic equation we get the answers of 4 and 12 seconds.

Instead of using the quadratic formula, you can just factor it, since 48 has 4 and 12 as factors and 4 + 12 = 16:
 t² - 16t + 48 = (t-4)(t-12) = 0
The quadratic equation will always work (assuming you use it properly) while factoring only work sometimes.

The projectile will be 768 ft from ground at t = 4s and t 12s, in the first when going up, secondly when going down.

A Pre-Calculus teacher would be expecting them to use a graphing utility to find the local max and local min (for the cubic function). The absolute value one can be taken care of using transformations to find the vertex.

Most schools provide graphing calculators. Students can also find graphing utilities online.

Roy, is this an online course or regular school? Either way I would caution against soliciting worked out solutions to problems that could be for graded assignments. For virtual schools in particular, students can copy & paste these answers directly into their assignment, which may even be a final exam. This creates an academic integrity issue that the tutor did not necessarily intend.

Do graphing calculators actually do algebraic solutions now and give absolute mins/maxes now or provide answers with radicals?

You have no idea. 

For Pre-Calculus, the teacher would expect rounded decimals in the intervals, which can be easily derived from a cheap graphing utility. However, in answer to your question, yes modern technology has the ability to generate solutions with radicals or variables. For Texas Instruments calculators, it would have a "CAS" designation, which stands for Computerized Algebra System. These types of calculators are allowed on both the SATs and the AP exams.

Most of what we used to do in math class is done on a calculator now. The focus of education has switched to applications and technology, rather than theory and procedure. I'm still old school, so I try to have a bit of both. We need students to know how the technology gets its answers, so they can use appropriate problem-solving techniques when technology fails. However, we also need to teach them how to use technology to their advantage, otherwise we are handicapping them in a digital world. That balance is hard to achieve, though, especially with standardized testing.

Most of what we used to do in math class is done on a calculator now. The focus of education has switched to applications and technology, rather than theory and procedure. I'm still old school, so I try to have a bit of both. We need students to know how the technology gets its answers, so they can use appropriate problem-solving techniques when technology fails. However, we also need to teach them how to use technology to their advantage, otherwise we are handicapping them in a digital world. That balance is hard to achieve, though, especially with standardized testing.

This reminded me of something my Second Year Circuits Prof said, it went something like: One day your calculator will give you an answer and it will be wrong and you will need to know why.

This reminds me something one of the TAs told me after I was having trouble with my code: "If Matlab isn't smart enough to solve [the differential equation] try Mathematica."

Then again, I'm pretty terrible at theory. For multivariable we were doing delta epsilon proofs. I do not know what those are, but I did them via memorization  .

I always prefer to do things myself, rather than plugging them into a computer program, although in the great Physicist tradition, I tend to look up any integral I can't do automatically or by a simple substitution since I was too lazy to do Integration by parts, and never remembered the other methods.

I don't remember exactly what delta epsilon proofs are, but I am fairly sure I have done them (or at least seen them done) my classes were mostly physics, so doing math proofs wasn't terribly common.

Roy, I PM'd you my cell phone # in case you have any future math questions. I currently teach Pre-Calculus, but I have taught up to AP Calculus AB.

By the way, when are you coming to visit? I think you promised about 5 years ago.