# Thread: determine all axis intercepts and asymptotes

1. ## determine all axis intercepts and asymptotes

how do you find the axis intercepts and asymptotes

f(x)=(3x^2-2x)/(x^2-4)

2. Originally Posted by oneway1225
how do you find the axis intercepts and asymptotes

f(x)=(3x^2-2x)/(x^2-4)

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

$f(0) = 0$ ... y-intercept

$f(x) = 0$ at $x = 0$ and at x = \frac{2}{3} ... x-intercepts

vertical asymptote, $x = -2$ and x = 2 ... $(x+2)(x-2)$ in the denominator

horizontal asymptote at $y = 3$ ... degree of the numerator = degree of the denominator ... ratio of the leading coefficients in the numerator and denominator gives you the value of the horizontal asymptote.

3. Originally Posted by oneway1225
how do you find the axis intercepts and asymptotes

f(x)=(3x^2-2x)/(x^2-4)
First note that $x \neq \pm 2$

X intercept:

$0 = \frac{3x^2 - 2x}{x^2 - 4}$

$0 = 3x^2 - 2x$

$0 = x(3x - 2)$

$x = \left\{0, \frac{2}{3}\right\}$.

Y intercept:

$y = \frac{3(0)^2 - 2(0)}{0^2 - 4}$

$y = \frac{0}{-4}$

$y = 0$.

Asymptotes:

$x = \pm 2$

4. Originally Posted by skeeter
$f(x) = \frac{3x(x-2)}{(x+2)(x-2)}$

$f(0) = 0$ ... y-intercept

$f(x) = 0$ at $x = 0$ ... x-intercept

vertical asymptote, $x = -2$ ... $(x+2)$ in the denominator

removable discontinuity (a "hole") at $x = 2$ ... $(x-2)$ in both numerator and denominator

horizontal asymptote at $y = 3$ ... degree of the numerator = degree of the denominator ... ratio of the leading coefficients in the numerator and denominator gives you the value of the horizontal asymptote.
The numerator is $3x^2-2x$ or $x(3x-2)$, not $3x^2-6x$, so we have 2 vertical asymptotes and no hole. The other vertical asymptote is at $x=2$

5. Originally Posted by oneway1225
how do you find the axis intercepts and asymptotes
To learn the general rules, study some online lessons on asymptotes and intercepts. Then please return here and review the solutions you were provided for this particular exercise, and confirm that you understand the concepts and methods they used.