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{{ printedBook.courseTrack.name }} {{ printedBook.name }} To graph the given rational function, we will find its domain, asymptotes and intercepts. Then, we will find points using a table of values. Finally, we will plot and connect those points.

Consider the given function.
$y=x−43x $
Recall that division by zero is not defined. Therefore, the rational function is undefined where $x−4=0.$
$x−4=0⇔x=4 $
The above means that $x=4$ is **not** included in the domain.
$Domain All real numbers exceptx=4. $

Asymptotes can be vertical or horizontal lines.

Consider the given function.
$y=x−43x $
Note that we cannot cancel out common factors. Therefore, there are no holes

and if a real number $a$ is **not** included in the domain, there is a vertical asymptote at $x=a.$ In this case, we have a vertical asymptote at $x=4.$

Let's pay close attention to the degrees of the numerator and denominator. $y=x_{1}−43x_{1} $ We see that the degrees of the numerator and denominator are the same. To find the horizontal asymptote, we need to find the quotient between the leading coefficients. $y=1x_{1}−43x_{1} $ Since $13 =3,$ there is a horizontal asymptote at $y=3.$

The intercepts of the function are the points at which the graph intersects the axes.

$y=x−43x $

Substitute$x=0$

$y=0−43(0) $

Solve for $x$

ZeroPropMultZero Property of Multiplication

$y=0−40 $

SubTermSubtract term

$y=-40 $

CalcQuotCalculate quotient

$y=0$

Let's make a table of values to graph the given function. Make sure to only use values included in the domain of the function.

$x$ | $x−43x $ | $y=x−43x $ |
---|---|---|

$-6$ | $-6−43(-6) $ | $1.8$ |

$-2$ | $-2−43(-2) $ | $1$ |

$2$ | $2−43(2) $ | $-3$ |

$6$ | $6−43(6) $ | $9$ |

$8$ | $8−43(8) $ | $6$ |

$10$ | $10−43(10) $ | $5$ |

Finally, let's plot and connect the points. Do not forget to draw the asymptotes and to plot the intercepts.