
Title:
Graph of Cubic Function  College Algebra

Description:

We just talked about the parent function for a cubic functions, f of x equals

xcubed. So I thought I should show you the graph of this function. Here it is,

this blue, pretty curve. You can see that this has the same overall behavior, as

the other cubic function that we saw in the last example. That one looked a

little more like this. You can see that the overall behavior is the same. One

end of the graph is going down to negative infinity, and the other end of the

graph is going up to positive infinity. In the middle, there's sort of a grey

area of, in this case, going down and then going up again, and in this case,

sort of leveling out for a bit. Now as I said in the last quiz, this parent

function looks really different from the parent function for quadratic

functions. Let's just add that on to our graph, so we can compare them more

easily. So here, once again, is the parent function for a quadratic function, f

of x equals x squared. Let's examine the overall behavior of this graph as well.

We know that the general shape of a parabola is kind of like a u, or if it's

upside down, like an n. Whether it's opening upward or opening downward, the

parabola has a vertex, which is either its minimum or its maximum. And then both

of its ends point in the same direction. And then as x gets further away in

either the negative direction,or the positive direction, the graph points the

same way. Either both ends of it go to positive infinity, or both ends go to

negative infinity. We also talked earlier about how a parabola has an axis of

symmetry running down the middle of it. So that if you fold the graph in half

along that line, it will exactly map to the other side of itself. That's not the

case over here with x cubed. If we simply fold this graph in half, down the

yaxis, this right side is not going to end up looking just like the left side.

It'll look more like this, not the same. This is all really interesting stuff.

Now since we're increasing powers, why don't we just do that one more time. So

I've put both y equals x squared and y equals x cubed on the same coordinate

plane, then I draw another graph over here for you. My question for you now is

what function does this graph represent? Now think about what this graph does on

either side of the y axis. And also think about what points you'd expect each of

these functions to go through. Your choices are y equals x, y equals x squared,

y equals x cubed, y equals x to the fourth and y equals x to the fifth. So you

can always make a t chart and plug in some points and see which of those t

charts best matches this graph.