Exercise 31 Page 521

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.

Domain

Consider the given function. y=x+4/x-4 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 except x=4

Asymptotes

Asymptotes can be vertical or horizontal lines.

Vertical Asymptote

Consider the given function. y=x+4/x-4 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.

Horizontal Asymptote

Let's pay close attention to the degrees of the numerator and denominator. y=x^1+4/x^1-4 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+4/1x^1-4 Since 11=1, there is a horizontal asymptote at y=1.

Intercepts

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

x-intercepts

The x-intercepts are the points where the graph intersects the x-axis. At these points, the value of the y-coordinate is zero. Let's substitute 0 for y in the given function and solve for x.

There is an x-intercept at (-4,0).

y-intercept

The y-intercept is the point where the graph intersects the y-axis. At this point, the value of the x-coordinate is zero. Let's substitute 0 for x in the given function and solve for y.

There is a y-intercept at (0,-1).

Graph

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.

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