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
$f(x) = 2 + \frac{5}{x - 3}$
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

A vertical asymptote occurs where the denominator is zero (and numerator non - zero). For \(f(x)=2+\frac{5}{x - 3}=\frac{2(x - 3)+5}{x - 3}=\frac{2x-6 + 5}{x - 3}=\frac{2x-1}{x - 3}\), the denominator \(x - 3=0\) when \(x = 3\). The numerator at \(x = 3\) is \(2(3)-1=5
eq0\), so vertical asymptote is \(x = 3\).

Step2: Find Horizontal Asymptote

For a rational function \(y=\frac{ax^n+...}{bx^m+...}\), if \(n=m\), horizontal asymptote is \(y=\frac{a}{b}\). Here, \(f(x)=\frac{2x - 1}{x - 3}\), the degree of numerator and denominator is 1. So horizontal asymptote is \(y=\frac{2}{1}=2\).

Step3: Find x - intercept

Set \(y = 0\), so \(0=2+\frac{5}{x - 3}\). Subtract 2: \(-2=\frac{5}{x - 3}\). Cross - multiply: \(-2(x - 3)=5\). Expand: \(-2x+6 = 5\). Subtract 6: \(-2x=-1\). Divide by - 2: \(x=\frac{1}{2}=0.5\). So x - intercept is \((0.5,0)\).

Step4: Find y - intercept

Set \(x = 0\), \(f(0)=2+\frac{5}{0 - 3}=2-\frac{5}{3}=\frac{6 - 5}{3}=\frac{1}{3}\approx0.333\). So y - intercept is \((0,\frac{1}{3})\).

Answer:

• Vertical Asymptote: \(x = 3\)
• Horizontal Asymptote: \(y = 2\)
• x - intercept: \((\frac{1}{2},0)\) (or \((0.5,0)\))
• y - intercept: \((0,\frac{1}{3})\) (or \((0,0.333)\))