Question

determine the vertical asymptotes of the graph of the function. give your answers as equations in exact form.

( h ( a ) = \frac { a + 1 } { 3 a ^ { 2 } - 7 a + 2 } )

separate multiple equations with commas as necessary. select
one\ if applicable.

equation(s) of the vertical asymptote(s):

Explanation:

Step1: Factor the denominator

Factor \(3a^{2}-7a + 2\). Using the formula \(ax^{2}+bx + c=a(x - x_1)(x - x_2)\) where \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for \(3a^{2}-7a + 2\) (\(a = 3\), \(b=-7\), \(c = 2\)), we have \(a=\frac{7\pm\sqrt{49-24}}{6}=\frac{7\pm5}{6}\). So \(3a^{2}-7a + 2=(3a - 1)(a - 2)\).

Step2: Find the vertical asymptotes

The vertical asymptotes of a rational function \(y=\frac{f(a)}{g(a)}\) occur at the values of \(a\) that make \(g(a)=0\) (provided \(f(a)
eq0\) at those values). Set \(3a - 1=0\) gives \(a=\frac{1}{3}\), and set \(a - 2=0\) gives \(a = 2\). Also, check \(h(\frac{1}{3})=\frac{\frac{1}{3}+1}{0}\) (undefined) and \(h(2)=\frac{2 + 1}{0}\) (undefined) and \(a+1
eq0\) when \(a=\frac{1}{3}\) or \(a = 2\).

Answer:

\(a=\frac{1}{3},a = 2\)