Question

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 pos
$f(x) = 1 + \frac{3}{x + 6}$
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 \( f(x)=1+\frac{3}{x + 6}=\frac{x + 6+3}{x + 6}=\frac{x + 9}{x + 6} \), set denominator \( x + 6=0 \).
\( x=-6 \)

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, numerator \( x + 9 \) (degree 1) and denominator \( x + 6 \) (degree 1). Leading coefficients are both 1, so horizontal asymptote is \( y = 1 \).

Step3: Find x - intercept

Set \( f(x)=0 \), so \( \frac{x + 9}{x + 6}=0 \). A fraction is zero when numerator is zero (denominator non - zero).
\( x+9 = 0\Rightarrow x=-9 \)

Step4: Find y - intercept

Set \( x = 0 \) in \( f(x) \). \( f(0)=1+\frac{3}{0 + 6}=1+\frac{3}{6}=1+\frac{1}{2}=\frac{3}{2}=1.5 \)

Answer:

• Vertical Asymptote: \( x=-6 \)
• Horizontal Asymptote: \( y = 1 \)
• x - intercept: \( x=-9 \) (point \((-9,0)\))
• y - intercept: \( y = 1.5\) (point \((0,1.5)\))