Questions: Find the vertical, horizontal, and oblique asymptotes, if any, of the function Q(x)=(3x^2-10x-8)/(5x^2-19x-4) Find the vertical asymptotes. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function has one vertical asymptote, . (Type an equation. Use integers or fractions for any numbers in your answer.) B. The function has two vertical asymptotes. The leftmost asymptote is . (Type equations. Use integers or fractions for any numbers in your answers.) C. The function has no vertical asymptote.

Find the vertical, horizontal, and oblique asymptotes, if any, of the function Q(x)=(3x^2-10x-8)/(5x^2-19x-4) Find the vertical asymptotes. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function has one vertical asymptote, . (Type an equation. Use integers or fractions for any numbers in your answer.) B. The function has two vertical asymptotes. The leftmost asymptote is . (Type equations. Use integers or fractions for any numbers in your answers.) C. The function has no vertical asymptote.
Transcript text: Find the vertical, horizontal, and oblique asymptotes, if any, of the function \[ Q(x)=\frac{3 x^{2}-10 x-8}{5 x^{2}-19 x-4} \] Find the vertical asymptotes. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function has one vertical asymptote, $\square$. (Type an equation. Use integers or fractions for any numbers in your answer.) B. The function has two vertical asymptotes. The leftmost asymptote is $\square$. (Type equations. Use integers or fractions for any numbers in your answers.) C. The function has no vertical asymptote.
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Solution

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Solution Steps

To find the vertical asymptotes of the rational function \( Q(x) = \frac{3x^2 - 10x - 8}{5x^2 - 19x - 4} \), we need to determine the values of \( x \) that make the denominator zero, as these are the points where the function is undefined and potentially has vertical asymptotes.

Solution Approach
  1. Set the denominator equal to zero and solve for \( x \).
  2. The solutions to this equation are the vertical asymptotes.
Step 1: Identify the Function

We are given the rational function

\[ Q(x) = \frac{3x^2 - 10x - 8}{5x^2 - 19x - 4}. \]

Step 2: Find the Denominator

To find the vertical asymptotes, we need to set the denominator equal to zero:

\[ 5x^2 - 19x - 4 = 0. \]

Step 3: Solve for Vertical Asymptotes

Solving the equation \(5x^2 - 19x - 4 = 0\) yields the vertical asymptotes at:

\[ x = -\frac{1}{5} \quad \text{and} \quad x = 4. \]

Final Answer

The function has two vertical asymptotes at

\[ \boxed{x = -\frac{1}{5}} \quad \text{and} \quad \boxed{x = 4}. \]

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