Name: Graph of Rational Functions
Uploaded: 2019-07-19
Duration: 54 min 4 s
Description: In this video, we will learn the concept of graphing the rational functions and the number of asymptotes in that graph in detail, using various examples.

Question

How can understanding transformations of rational functions help you graph them?

Accepted Answer

Without understanding the transformations, we have only one method to plot the rational functions which is finding the points that lies on the graph from the given function, followed by plotting them on the $x y -$ plane and then joining those points.

This method required lot of calculation and consumes so much time of a student. But knowing the transformation can reduce the time for graphing the given function.

For Example: Let us sketch the graph of the function, $g (x) = \frac{10}{x - 3} + 2$ using the graph of the function, $f (x) = \frac{1}{x}$

Solution: The given function, can be written as $g (x) = 10 f (x - 3) + 2$ , where $f (x) = \frac{1}{x}$

Thus, to obtain the graph of $g (x)$ , we need to translate the graph of $f (x)$ horizontally by $3$ units.

Such that we obtain the graph of $\frac{1}{x - 3}$ .

Now, we stretch the graph of $\frac{1}{x - 3}$ vertically by scale factor $10$ to obtain the graph of $\frac{10}{x - 3}$ .

Further, we shift the graph of $\frac{10}{x - 3}$ vertically up by $2$ units.

Now, we find the asymptotes for the given function,

Now, as $x \to \infty, g (x) \to 2$

Thus, horizontal asymptote is $g (x) = 2$

For vertical asymptote,

We know that as $x \to 3$ , $g (x) \to \infty$

Thus, $x = 3$ is a vertical asymptote

Hence, the graph of the function $g (x) = \frac{10}{x - 3} + 2$ is given as follows:

How can understanding transformations of rational functions help you graph them?

Important Questions on Graphs of Rational Functions

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