Questions: Find the horizontal and vertical asymptotes of the curve. You may want to use a graphing calculator (or computer) to check your work by graphing the curve and estimating the asymptotes. (Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.) y=(2x^2+9)/(5x^2+34x-7) x= y=

Find the horizontal and vertical asymptotes of the curve. You may want to use a graphing calculator (or computer) to check your work by graphing the curve and estimating the asymptotes. (Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.) y=(2x^2+9)/(5x^2+34x-7) x= y=
Transcript text: Find the horizontal and vertical asymptotes of the curve. You may want to use a graphing calculator (or computer) to cneck your work by graphing the curve and estimating the asymptotes. (Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.) \[ \begin{array}{l} y=\frac{2 x^{2}+9}{5 x^{2}+34 x-7} \\ x=\square \\ y=\square \end{array} \]
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Solution

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Solution Steps

Step 1: Finding Horizontal Asymptotes

To find the horizontal asymptotes of the function \( y = \frac{2x^2 + 9}{5x^2 + 34x - 7} \), we analyze the behavior of the function as \( x \) approaches infinity and negative infinity.

Calculating the limits: \[ \lim_{x \to \infty} y = \frac{2}{5}, \quad \lim_{x \to -\infty} y = \frac{2}{5} \] Thus, the horizontal asymptote is \( y = \frac{2}{5} \).

Step 2: Finding Vertical Asymptotes

To find the vertical asymptotes, we set the denominator equal to zero: \[ 5x^2 + 34x - 7 = 0 \] Solving this quadratic equation gives us the potential vertical asymptotes: \[ x = -7, \quad x = \frac{1}{5} \] Next, we check if these values also make the numerator zero: \[ 2(-7)^2 + 9 \neq 0, \quad 2\left(\frac{1}{5}\right)^2 + 9 \neq 0 \] Since neither value zeroes out the numerator, both are valid vertical asymptotes.

Final Answer

The horizontal and vertical asymptotes of the curve are: \[ \text{Horizontal Asymptote: } y = \frac{2}{5} \] \[ \text{Vertical Asymptotes: } x = -7, \frac{1}{5} \] Thus, the final answer is: \[ \boxed{y = \frac{2}{5}, x = -7, \frac{1}{5}} \]

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