Question

plot all of the existing five features of the following rational function (some may not be needed). if you get a fraction or decimal then plot as close to the true location as possible.
$f(x) = \frac{-5x}{3x - 11}$
plot rational function
vertical asymptote horizontal asymptote
x-intercept y-intercept hole
click on a feature then drag it into place.

Explanation:

Step1: Find Vertical Asymptote

Set denominator equal to zero: \(3x - 11 = 0\). Solve for \(x\): \(3x=11\) → \(x = \frac{11}{3}\approx3.67\).

Step2: Find Horizontal Asymptote

For rational function \(\frac{ax + b}{cx + d}\), horizontal asymptote is \(\frac{a}{c}\) when degrees are equal. Here, \(a=-5\), \(c = 3\), so \(y=-\frac{5}{3}\approx - 1.67\).

Step3: Find x - Intercept

Set numerator equal to zero: \(-5x=0\) → \(x = 0\). So x - intercept is \((0,0)\).

Step4: Find y - Intercept

Set \(x = 0\) in \(f(x)\): \(f(0)=\frac{-5(0)}{3(0)-11}=0\). So y - intercept is \((0,0)\).

Step5: Check for Hole

Hole occurs when numerator and denominator have common factor. Here, numerator is \(-5x\), denominator is \(3x - 11\), no common factors, so no hole.

Answer:

• Vertical Asymptote: \(x=\frac{11}{3}\) (or \(x\approx3.67\))
• Horizontal Asymptote: \(y =-\frac{5}{3}\) (or \(y\approx - 1.67\))
• x - Intercept: \((0,0)\)
• y - Intercept: \((0,0)\)
• Hole: None