What's the shape of the graph y = 1/x + x^2 ?

Question

Is it a parabola? because there is x^2?
Or is it something else because of the 1/x?

Answer 1

It's something else. I don't know of a good way to get the exact shape without plugging in values for x and y and just plotting key points. One important thing to look at is the fact that the function is not an even function, whereas the function y=x^2 is an even function. A function is "even" if it is symmetric across the y axis.

y=x^2 takes on positive values for all values of x because it is squared.

y=1/x + x^2 will take on negative values for some values of x and positive values for other values of x. Also, the function gets really big for really small fractions. If you plug in .01 for x, you get 100.0001.

If you plug .01 into x^2 you get .0001.

I would just try plotting a bunch of values and seeing what it does.

Answer 2

A little bit of everything, Bob.
The graph of y = 1/x is a hyperbola with branches in the first and third quadrant. If you look at the first quadrant, at large x, the hyperbola contribution is very small, and the graph is like the parabola. Some where around x=2, as the parabola contribution continues to decrease, the hyperbolic contribution starts to take over. So rather than having a curve that flattens out and reaches the origin, the curve starts to increase rapidly as x approaches 0 from the right. You can take the minimum of the curve, which occurs at
x = cube root(1/2)

Answer 3

THink of it this way.

As x is close to zero, x^2 is small while 1/x is large (in magnitude). So the 1/x dominates.

When x is large, the x^2 dominates.

So it's a kind of combo between the two.
There obviously will be a local minimum in the positive x scale and a point of inflexion on the negative x scale.

Answer 4

Yes, it is "something else". The 1/x is hyperbolic and adds with the x*2 parabola so you have two figures filling the first and third quadrants.

Answer 5

It's something else. As x goes to zero, 1/x becomes extremely large. Therefore, from x=0 to x=1, y is huge. Also, as x goes to zero from the left, y becomes extremely small.