Question

student activity sheet 5; exploring \asymptotes and removable discontinuities\ page 4 of 11

1. reinforce identify the vertical and horizontal asymptotes of each rational function.

a. ( q(x)=\frac{8 x-3}{4 x+8} )
b. ( p(x)=\frac{-9 x+3}{3 x-18} )
c. ( f(x)=\frac{2 x-7}{5 x+15} )
d. ( h(x)=\frac{\frac{1}{2} x+2}{-2 x+8} )

Explanation:

Step1: Find vertical asymptote

For a rational function \(y = \frac{f(x)}{g(x)}\), vertical asymptote is found by setting \(g(x)=0\).

a.

For \(q(x)=\frac{8x - 3}{4x + 8}\), set \(4x+8 = 0\).
\(4x=-8\), so \(x=-2\).

b.

For \(p(x)=\frac{-9x + 3}{3x - 18}\), set \(3x-18 = 0\).
\(3x=18\), so \(x = 6\).

c.

For \(f(x)=\frac{2x - 7}{5x + 15}\), set \(5x+15=0\).
\(5x=-15\), so \(x=-3\).

d.

For \(h(x)=\frac{\frac{1}{2}x + 2}{-2x + 8}\), set \(-2x + 8=0\).
\(-2x=-8\), so \(x = 4\).

Step2: Find horizontal asymptote

For a rational function \(y=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots+b_0}\), if \(n = m\), horizontal asymptote is \(y=\frac{a_n}{b_m}\).

a.

For \(q(x)=\frac{8x - 3}{4x + 8}\), \(n = m = 1\), \(a_n=8\), \(b_m=4\).
Horizontal asymptote: \(y=\frac{8}{4}=2\).

b.

For \(p(x)=\frac{-9x + 3}{3x - 18}\), \(n = m = 1\), \(a_n=-9\), \(b_m=3\).
Horizontal asymptote: \(y=\frac{-9}{3}=-3\).

c.

For \(f(x)=\frac{2x - 7}{5x + 15}\), \(n = m = 1\), \(a_n=2\), \(b_m=5\).
Horizontal asymptote: \(y=\frac{2}{5}\).

d.

For \(h(x)=\frac{\frac{1}{2}x + 2}{-2x + 8}\), \(n = m = 1\), \(a_n=\frac{1}{2}\), \(b_m=-2\).
Horizontal asymptote: \(y=\frac{\frac{1}{2}}{-2}=-\frac{1}{4}\).

Answer:

a. Vertical asymptote: \(x=-2\), Horizontal asymptote: \(y = 2\)
b. Vertical asymptote: \(x = 6\), Horizontal asymptote: \(y=-3\)
c. Vertical asymptote: \(x=-3\), Horizontal asymptote: \(y=\frac{2}{5}\)
d. Vertical asymptote: \(x = 4\), Horizontal asymptote: \(y=-\frac{1}{4}\)