Questions: Graphing Asymptotes for a Rational Functions Two copies of the same Rational Function are shown below. On the graph below draw the Horizontal Asymptote and write the equation for the horizontal asymptote underneath. f(x) = 2x / (x + 3) Horizontal Asymptote:

Solution Steps

The given rational function is \( f(z) = \frac{2z}{z + 3} \).

For a rational function of the form \( \frac{a_n z^n + \ldots}{b_m z^m + \ldots} \):

• If \( n < m \), the horizontal asymptote is \( y = 0 \).
• If \( n = m \), the horizontal asymptote is \( y = \frac{a_n}{b_m} \).
• If \( n > m \), there is no horizontal asymptote.

In this case, the degrees of the numerator and denominator are both 1 (since the highest power of \( z \) in both the numerator and the denominator is 1). Therefore, the horizontal asymptote is \( y = \frac{2}{1} = 2 \).

The equation of the horizontal asymptote is \( y = 2 \).

Final Answer