Parabola problem help!!

Question

While training for an upcoming event , Mary Jo made a dive whose path is described by the function f(x) = 5x^2/2 -10x+6 .

A) from what height did she dive
B) how long did it take Mary Jo to reach the lowest point in her dive?
C) how long did she remain underwater?
D) how long did her dive last?
E) determine the domaine and range of this function

Answer 1

I think Michael was picturing a dive from a board going up then down in a parabola, but this question looks like it's ok as written.  (Just when you write the fraction use parentheses to show whats on top and bottom, like (5x^2)/2 shows that the 5x^2 is all on top and the 2 is on bottom, or (5/2)x^2 shows that the fraction is multiplying the x^2 and not just the 2.)

 

The height she dives from is her starting point, which is when x = 0 (time of 0 seconds).  This should be 6 (feet?  meters?).

 

The other answers can be found in various ways, the easiest of which is to graph your function and simply look at the parts you care about.  Obviously, depending on your teacher and your available calculator, this may or may not be acceptable.  But even if you need to show work, it's still a good thing to do to get an idea of what the answers will be before doing your calculations.

 

According to desmos.com/calculator (love that graphing calc), the lowest point is at (2,-4), which means it took her 2 seconds to get to the lowest point, and she reached a depth of 4 (feet, meters, something).  Without a calculator you could use the equation for the axis of symmetry x = -b/(2a) where b = -10 and a = (5/2).

 

Her time underwater is like the graph being under the x axis, which means we need the roots or zeroes of the parabola.  Normally you can factor, but with the fraction that's a little tougher to do (still possible if you want to try though).  I'd say use the quadratic formula for this one and hopefully you get roots of .735 and 3.625 (read from graph).  The time between those is the time she spends underwater.

 

I'm going to assume her dive ends once she surfaces at (3.625,0), so it lasted 3.625 seconds.

 

The domain (leftmost value to rightmost value) for any parabola will be all real numbers since we can plug in any x value we want to get a y value.  This is shown on the graph as the function extending to the left and right infinitely.  The range is the lowest point to the highest point, and since her lowest depth was -4, the range is from -4 up to infinity (possibly written as y >= -4).  Really, the range of her dive was from -4 to 6, but it says the function, so I'm just going based on the whole graph.

 

Hope that helps!