# How do you graph y=4/(3x-6)+5 using asymptotes, intercepts, end behavior?

Dec 20, 2016

graph{(4/(3x-6))+5 [-17.58, 22.42, -6.32, 13.68]}

asymptote: $x = 2$, vertical

$y$-intercept is at $\left(0 , 4.33\right)$

$x$-intercept is at $\left(1.73 , 0\right)$

$y < 5$ when $x < 2$
$y > 5$ when $x > 2$

#### Explanation:

$\frac{n}{0}$ = undefined

$\frac{n}{0} + 5$ =undefined

asymptote: $\frac{4}{3 x - 6} = \frac{4}{0}$

$3 x - 6 = 0$

$3 x = 6$

$x = 2$

intercepts:

$y$:
$y$-intercept: $x = 0$

$y = \frac{4}{3 x - 6} + 5$

$= \frac{4}{-} 6 + 5$

$= 4 \frac{1}{3} \mathmr{and} 4.33 \left(3 s . f .\right)$

$x$:
$x$-intercept: $y = 0$

$\frac{4}{3 x - 6} + 5 = 0$

$\frac{4}{3 x - 6} = - 5$

$3 x - 6 = - 0.8$

$3 x = - 0.8 + 6 = 5.2$

$x = \frac{5.2}{3} = 1.73 \left(3 s . f .\right)$

end behaviour:

$y < 5$ when $x < 2$

e.g. $x = 1.5$:

$\frac{4}{3 x - 6} + 5 = \frac{4}{-} 1.5 + 5$

$= \frac{8}{3} + 5$

$= 7 \frac{2}{3}$

$y > 5$ when $x > 2$

e.g. $x = 2.1$:

$\frac{4}{3 x - 6} + 5 = \frac{4}{0.3} + 5$

$= \frac{40}{3} + 5$

$= 18 \frac{1}{3}$