Question

draw the lines representing the vertical and horizontal asymptotes on the graph of this function.

\\f(x) = \frac{2x^2 + 7x - 4}{x^2 + 5x + 4}\\

Explanation:

Simplify the rational function

$$ f(x) = \frac{2x^2 + 7x - 4}{x^2 + 5x + 4} = \frac{(2x - 1)(x + 4)}{(x + 1)(x + 4)} $$

For \(x
eq -4\):

$$ f(x) = \frac{2x - 1}{x + 1} $$

Find the vertical asymptote

The denominator of the simplified function is zero when:

$$ x + 1 = 0 \implies x = -1 $$

Thus, the vertical asymptote is the line \(x = -1\).

Find the horizontal asymptote

Since the degree of the numerator equals the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients:

$$ y = \frac{2}{1} \implies y = 2 $$

Answer:

The asymptotes of the function \(f(x) = \frac{2x^2 + 7x - 4}{x^2 + 5x + 4}\) are:

• Vertical Asymptote: \(x = -1\)
• Horizontal Asymptote: \(y = 2\)