y = (4x) / (x^2+4)?

Question

so i have to manually graph this function... its easy except im kinda confused about one thing... it has an x- and y- intercept at (0, 0), but it also has a horizontal asymptote at y=0... how does this make sence? it cannot mean that there is a point discontinuity since there is a value for x and y at 0, without having to set limits...

Answer 1

There are no discontinuities since the denominator never gets below +4 and no special functions are used (e.g., logarithms). The function is continuous everywhere.

It will have a point (0,0). If you check a few values, you will find that it appears continuous in the neighbourhood of (0,0).

The denominator is always positive (and never less than +4), while the numerator will be positive for x>0 and negative for x<0. Therefore, the function cannot stay at zero when x moves away from zero.

Because the denominator is a degree higher than the numerator, the denominator will grow a lot faster than the numerator so that Y will tend to zero as x grows to infinity (or to minus infinity).

You can also note that the function is symmetrical about (0,0) -- point symmetry. In other words Y(x) = -Y(-x).

In light of everything above, there must be some maximum value to the right of (0,0) and a minimum value to the left of x (and both must be at the same absolute value of x).

You can find the exact value by differentiating the equation (finding dY/dx for a division).

But your question is simply: is this possible?

So let us check: are there values of x that allow us to have Y not equal to zero?

Let us try Y = 1

1 = (4x) / (x^2+4)
x^2 + 4 = 4x
x^2 - 4x + 4 = 0
x = [ 4 +/- SQRT(16-16)]/2
x = 2
(and, by symmetry, Y= -1 when x = -2)

Therefore, it is possible for Y to get at least as far as 1.

Looking at positive values of x, here are some points (values rounded):

(0,0)
(1, 0.8)
(2, 1)
(3, 0.923)
(4, 0.8)
(10, 0.385)
(100, 0.04)
(1000, 0.004)
(1,000,000, 0.000004)

Answer 2

There it no H.A. at zero. Set y=0 and multiply by x^2+4. You then get 0=4x. Divide 0=4x by 4 so you get (0/4)=x, and when you have zero fourths of a pie of pizza, you have none, zero zero works.

Answer 3

A horizontal asymptote represents end behaviors, not at the origin.