I need someone to explain this to me. I'm am mainly confused on HOW to graph the function after I have factored.
The volume, V, in cubic centimeters, of a collection of open-topped boxes can be modelled by
V(x) = 4x^3 - 220x^2 + 2800x
where x is the height of each box in centimetres.
A) Graph V(x). State the restrictions.
*HOW do I graph this function?
(step1) set equal zero
(step2) factor: 0 = 4x (x - 20) (x - 35)
.....................zeros {0, 20, 35}
Now how do I graph this, I am confused?
B) Fully Factor V(x). State the relationship between the factored form of the equation and the graph
C) State the value of the constant finite differences for this function.
*Answers*
a) Graph.
....x>0 and V(x)>0
b)V(x) = 4x(x-35)(x-20)
As a practical matter, let's look at some "qualitative" aspects of the graph of V(x):
Fact 1: we know the graph is a cubic equation, which crosses the x-axis at x = 0, x = 20, x = 35
Fact 2: the leading term (4x^{3}) is positive, meaning V(x) tends to infinity for very large positive x, and to negative infinity for very large negative x.
It stands to reason that V(x) is mostly going "up" until it gets to some point between 0 and 20 (where it crosses the x-axis), and mostly going "down" until some point between 20 and 35 (where it crosses the x-axis again, never to return), and goes mostly up thereafter.
(I say "mostly" because I want to avoid the STRONGER statements we can say using the differential calculus).
Now, let's look at what the factor (x - 20)(x - 35) looks like:
It is a parabola opening "up", with a vertex at (55/2,k) for some k < 0. Plugging x = 55/2 into (x - 20)(x - 35) gives us k = -225/4 (although this isn't terribly helpful).
Now what happens to this parabola, when we stretch it by a factor of 4x?
From (-∞,0) it turns the parabolic curve upside-down, and curves it a bit sharper. On [0,20], the curve is positive, but it can only get so high, because one factor (the 4x) is getting larger, but the other one ((x - 20)(x - 35)) is getting smaller.
Eventually the (x - 20)(x - 35) part brings it back down to 0 (at x = 20), and the graph goes negative. Now all this time 4x is still getting bigger, and eventually (at x = 55/2) the parabola stops going down, so after that, both factors start pulling together to increase the value of V(x).
Intuitively, since the factor 4x is "deforming" the parabola, one would expect the "peak" of the cubic we get to be different than 10, and the bottom of the "valley" to be different than 55/2 (but they shouldn't be terribly far away from these points).