Question

Explain how you can tell (without graphing it) that the rational function

r(x)= x^6 +10 / x^4+8x^2+15

has no x intercept and no horizontal, vertical, or slant asymptote. What is its end behaviour?

Please help? thankq

Answer 1

Horizontal/slant intercepts:
divide the leading coefficient of the numerator by that of the denominator:
q=x^6/x^4=x²
If q is a numerical constant, the horizontal asymptote is at y=q.

If q is a linear term, such as 2x, then there is a slant asymptote along the line y=2x.

Vertical asymptotes occur where the denominator becomes zero.

Substitute y=x² in the denominator and solve for the resulting quadratic where y=-3 or -5. Clearly the solutions for x in y=-3 or -5 are complex, therefore the denominator does not become zero, hence no vertical asymptote.