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) = \frac{1}{x + 5} + 3 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 \( f(x)=\frac{1}{x + 5}+3 \), set \( x + 5=0 \).
\( x=-5 \)

Step2: Find Horizontal Asymptote

For rational functions of the form \( y=\frac{a}{x - h}+k \), the horizontal asymptote is \( y = k \). Here, \( k = 3 \), so horizontal asymptote is \( y = 3 \).

Step3: Find x - intercept

Set \( f(x)=0 \):
\( 0=\frac{1}{x + 5}+3 \)
Subtract 3 from both sides: \( - 3=\frac{1}{x + 5} \)
Multiply both sides by \( x + 5 \): \( -3(x + 5)=1 \)
Expand: \( -3x-15 = 1 \)
Add 15 to both sides: \( -3x=16 \)
Divide by - 3: \( x=-\frac{16}{3}\approx - 5.333 \)

Step4: Find y - intercept

Set \( x = 0 \):
\( f(0)=\frac{1}{0 + 5}+3=\frac{1}{5}+3=\frac{1 + 15}{5}=\frac{16}{5}=3.2 \)

Answer:

• Vertical Asymptote: \( x=-5 \)
• Horizontal Asymptote: \( y = 3 \)
• x - intercept: \( x=-\frac{16}{3}\) (or approximately \( x\approx - 5.33 \))
• y - intercept: \( y=\frac{16}{5}\) (or \( y = 3.2 \))