Graph f(x)=(2x^2+x-3)/(4x^2-3x-1)

$f (x) = \frac{2 x^{2} + x - 3}{4 x^{2} - 3 x - 1}$

Step 1

Find where the expression $\frac{2 x + 3}{4 x + 1}$ is undefined.

$x = - \frac{1}{4}$

Step 2

Consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 3

Find $n$ and $m$ .

$n = 1$

$m = 1$

Step 4

Since $n = m$ , the horizontal asymptote is the line $y = \frac{a}{b}$ where $a = 2$ and $b = 4$ .

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

Step 5

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - \frac{1}{4}$

Horizontal Asymptotes: $y = \frac{1}{2}$

No Oblique Asymptotes

Step 7