Question

problem. 6 : consider the rational function $f(x) = \frac{x^2 - 9x + 20}{x^2 - 3x + 2}$. identify any vertical asymptotes. (note: input answers below in increasing values of $x$; ie if your answers were $x = -3$ and $x = 22$, then the left answer box would be $-3$ and the right would be $22$). $x = \square$ $x = \square$ problem. 6.1 : are there any holes? if so, give the coordinates of the hole. if not, enter
one\. ( $\square$, $\square$ ) problem. 6.1.1 : identify any horizontal asymptotes. if none, enter
one\. $y = \square$

Explanation:

Step1: Analyze degrees of numerator and denominator

The rational function is \( f(x)=\frac{x^2 - 9x + 20}{x^2 - 3x + 2} \). The degree of the numerator (highest power of \( x \)) is 2, and the degree of the denominator is also 2.

Step2: Find the ratio of leading coefficients

The leading coefficient of the numerator is 1 (coefficient of \( x^2 \) in \( x^2 - 9x + 20 \)), and the leading coefficient of the denominator is 1 (coefficient of \( x^2 \) in \( x^2 - 3x + 2 \)). The ratio of the leading coefficients is \( \frac{1}{1}=1 \).

Step3: Determine the horizontal asymptote

For a rational function where the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients. So, the horizontal asymptote is \( y = 1 \).

Answer:

\( y = 1 \)