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 possible.
$f(x) = 6 + \frac{5}{x + 5}$
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

The vertical asymptote of a rational function occurs where the denominator is zero (and the numerator is not zero there). For \( f(x) = 6+\frac{5}{x + 5}\), set the denominator \( x + 5=0\).
\( x+5 = 0\) gives \( x=-5\). So the vertical asymptote is \( x=-5\).

Step2: Find Horizontal Asymptote

For a rational function \( f(x)=\frac{N(x)}{D(x)}\), if the degrees of \( N(x)\) and \( D(x)\) are equal, the horizontal asymptote is the ratio of leading coefficients. If the degree of \( N(x)\) is less than \( D(x)\), horizontal asymptote is \( y = 0\). If the degree of \( N(x)\) is greater, there is no horizontal asymptote (but an oblique one). Here, rewrite \( f(x)\) as \( f(x)=\frac{6(x + 5)+5}{x + 5}=\frac{6x+30 + 5}{x + 5}=\frac{6x+35}{x + 5}\). The degree of numerator (\(6x + 35\)) and denominator (\(x + 5\)) is 1 (equal). The leading coefficient of numerator is 6, denominator is 1. So horizontal asymptote is \( y=\frac{6}{1}=6\).

Step3: Find x - intercept

To find the x - intercept, set \( f(x)=0\). So \( 6+\frac{5}{x + 5}=0\).
Subtract 6 from both sides: \(\frac{5}{x + 5}=-6\).
Multiply both sides by \( x + 5\) (assuming \( x
eq - 5\)): \(5=-6(x + 5)\).
Expand: \(5=-6x-30\).
Add 30 to both sides: \(35=-6x\).
Divide by - 6: \(x=-\frac{35}{6}\approx - 5.83\).

Step4: Find y - intercept

To find the y - intercept, set \( x = 0\) in \( f(x)\).
\( f(0)=6+\frac{5}{0 + 5}=6 + 1=7\). So the y - intercept is at \( (0,7)\).

Answer:

• Vertical Asymptote: \( x=-5\)
• Horizontal Asymptote: \( y = 6\)
• x - intercept: \( x=-\frac{35}{6}\) (or \(x\approx - 5.83\))
• y - intercept: \( y = 7\) (point \((0,7)\))