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) = -6 - \frac{1}{x + 5} \\) 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

A vertical asymptote occurs where the denominator is zero (and numerator non - zero). For \(f(x)=-6-\frac{1}{x + 5}\), the denominator is \(x + 5\). Set \(x+5=0\), we get \(x=-5\). So the vertical asymptote is \(x=-5\).

Step2: Find Horizontal Asymptote

For a rational function of the form \(y=\frac{a}{x - h}+k\) (after re - writing), the horizontal asymptote is \(y = k\). Rewrite \(f(x)=-6-\frac{1}{x + 5}\) as \(f(x)=-\frac{1}{x+5}-6\). Here \(k=-6\), so the horizontal asymptote is \(y=-6\).

Step3: Find x - intercept

To find the x - intercept, set \(y = f(x)=0\). So \(0=-6-\frac{1}{x + 5}\). Add 6 to both sides: \(6=-\frac{1}{x + 5}\). Cross - multiply: \(6(x + 5)=-1\). Expand: \(6x+30=-1\). Subtract 30 from both sides: \(6x=-31\). Divide by 6: \(x=-\frac{31}{6}\approx - 5.17\) (but we can keep it as a fraction for exactness, \(x =-\frac{31}{6}\)).

Step4: Find y - intercept

To find the y - intercept, set \(x = 0\). Then \(f(0)=-6-\frac{1}{0 + 5}=-6-\frac{1}{5}=-\frac{30 + 1}{5}=-\frac{31}{5}=-6.2\).

Answer:

• Vertical Asymptote: \(x=-5\)
• Horizontal Asymptote: \(y = - 6\)
• x - intercept: \(x=-\frac{31}{6}\) (or approximately \(x\approx - 5.17\))
• y - intercept: \(y=-\frac{31}{5}\) (or approximately \(y=-6.2\))