Question

How do you graph$y = \dfrac{1}{{x + 3}}$?

Answer 1

In order to draw the graph of the above rational function we have to first find the asymptotes of the rational function, by setting denominator equal to 0 to find the vertical asymptote and for the horizontal asymptote find the limit approach to $\infty$for the function .The graph to the equation will be a rectangular hyperbola made about these asymptotes.

Complete step by step solution:
We are given an rational function of the form $R(x) = \dfrac{{a{x^n}}}{{b{x^m}}}$ , here we are having variable $x$and $y$telling $y = \dfrac{1}{{x + 3}}$
Vertical asymptotes for rational functions are found by setting the denominator equivalent to 0. This additionally assists with finding the domain.
The domain can NOT contain that number! For this function,
$x + 3 \ne 0 \\\ x \ne - 3 \\\$
y is defined for $x \in ( - \infty , - 3) \cup ( - 3, + \infty )$
so $x = - 3$ is the condition of the vertical asymptote, and 0 should be avoided with regard to the domain: $x \in ( - \infty , - 3) \cup ( - 3, + \infty )$
Horizontal asymptotes are found by subbing in huge positive and negative qualities into the
function. $f(1000)$ or $f(1000000)$can assist with figuring out where the function "closes" are going.
For this situation, $\dfrac{1}{{1000 + 3}}\,or\,\dfrac{1}{{1000000 + 3}}$
will get very near 0. (this is known as a limit)
$\mathop {\lim }\limits_{x \to \infty } y = 0$

Therefore, our horizontal asymptote will be at $y = 0$.

The graph to the equation y is a rectangular hyperbola about the vertical and horizontal asymptotes.

Hence we’ve successfully plotted our graph of $y = \dfrac{1}{{x + 3}}$.

Additional Information:
Cartesian Plane: A Cartesian Plane is given its name by the French mathematician Rene Descartes ,who first used this plane in the field of mathematics .It is defined as the two mutually perpendicular the number line , the one which is horizontal is given the name x-axis and the one which is vertical is known as y-axis. With the help of these axis we can plot any point on this cartesian plane with the help of an ordered pair of numbers.

Note: 1.Draw the cartesian plane only with the help of a straight ruler and pencil to get the perfect and accurate results.

2.You can take any two points from the equation to plot the graph to the equation