Question

plot all intercepts and asymptotes of the following rational function. if you get a fraction or decimal then plot as close to the true location as possible.
$f(x) = -5 + \frac{1}{x - 1}$
plot rational function
vertical asymptote horizontal asymptote
x-intercept y-intercept
click on a feature then drag it into place.

Explanation:

Step1: Find Vertical Asymptote

The vertical asymptote of a rational function occurs where the denominator is zero (and numerator is not zero). For \( f(x)=-5+\frac{1}{x - 1}\), the denominator is \( x - 1\). Set \( x - 1=0\), so \( x = 1\).

Step2: Find Horizontal Asymptote

For rational functions, if the degrees of numerator and denominator are equal, the horizontal asymptote is the ratio of leading coefficients. Here, rewrite \( f(x)=\frac{-5(x - 1)+1}{x - 1}=\frac{-5x+5 + 1}{x - 1}=\frac{-5x+6}{x - 1}\). The degree of numerator and denominator is 1. The leading coefficient of numerator is -5, denominator is 1. But also, as \( x
ightarrow\pm\infty\), \( \frac{1}{x - 1}
ightarrow0\), so \( f(x)
ightarrow - 5\). Thus, horizontal asymptote is \( y=-5\).

Step3: Find x - intercept

Set \( f(x) = 0\): \( 0=-5+\frac{1}{x - 1}\). Then \( 5=\frac{1}{x - 1}\), so \( 5(x - 1)=1\), \( 5x-5 = 1\), \( 5x=6\), \( x=\frac{6}{5}=1.2\).

Step4: Find y - intercept

Set \( x = 0\): \( f(0)=-5+\frac{1}{0 - 1}=-5-1=-6\).

Answer:

• Vertical Asymptote: \( x = 1\)
• Horizontal Asymptote: \( y=-5\)
• x - intercept: \( x=\frac{6}{5}\) (or \( x = 1.2\))
• y - intercept: \( y=-6\)