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.
$f(x) = \frac{-5}{x - 3} + 2$
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 \( f(x)=\frac{-5}{x - 3}+2\), the denominator is \(x - 3\). Set \(x - 3=0\), so \(x = 3\).

Step2: Find Horizontal Asymptote

For rational functions, if the degrees of numerator and denominator are equal (after combining terms), the horizontal asymptote is the ratio of leading coefficients. First, combine \(f(x)=\frac{-5}{x - 3}+2=\frac{-5 + 2(x - 3)}{x - 3}=\frac{2x-6 - 5}{x - 3}=\frac{2x-11}{x - 3}\). The degree of numerator (1) and denominator (1) are equal. The leading coefficient of numerator is 2, denominator is 1. So horizontal asymptote is \(y = 2\).

Step3: Find x - intercept

Set \(f(x)=0\): \(\frac{-5}{x - 3}+2 = 0\). Subtract 2: \(\frac{-5}{x - 3}=-2\). Multiply both sides by \(x - 3\) (assuming \(x
eq3\)): \(-5=-2(x - 3)\). Expand: \(-5=-2x + 6\). Subtract 6: \(-11=-2x\). So \(x=\frac{11}{2}=5.5\).

Step4: Find y - intercept

Set \(x = 0\): \(f(0)=\frac{-5}{0 - 3}+2=\frac{5}{3}+2=\frac{5 + 6}{3}=\frac{11}{3}\approx3.67\).

Answer:

• Vertical Asymptote: \(x = 3\)
• Horizontal Asymptote: \(y = 2\)
• x - intercept: \(x=\frac{11}{2}\) (or \(5.5\))
• y - intercept: \(y=\frac{11}{3}\) (or approximately \(3.67\))