Question

Consider the following rational function: f(x)=(x^2+8x+7)/(x^2+11x+30).

Find the x and y-intercepts.

Factor the numerator and denominator using the trial and error method. are there any common factors between the numerator and denominator?

Find the vertical and horizontal asymptote

Answer 1

x²+8x+7=(x+1)(x+7), when x=0, this expression=7.

x²+11x+30=(x+6)(x+5), when x=0, this expression=30.

f(x)=(x+1)(x+7)/((x+6)(x+5)) (red).

The y-intercept=7/30 (0,0.2333).

The x-intercepts are x=-1 and -7 (zeroes of the numerator).

The vertical asymptotes (green) are x=-5 and -6 because these values make the denominator zero.

The horizontal asymptote (blue) is x²/x²=1, so it’s y=1, because, when x is very large (positive or negative), all terms apart from x² can be ignored as insignificant.