Question

score 4/10 penalty none 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{4}{x - 1} + 1 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 there). For \( f(x)=\frac{4}{x - 1}+1\), the denominator of the fractional part is \( x - 1\). Set \( x - 1=0\), so \( x = 1\).

Step2: Find Horizontal Asymptote

Rewrite \( f(x)\) as a single fraction: \( f(x)=\frac{4+(x - 1)}{x - 1}=\frac{x + 3}{x - 1}\). For rational functions \(\frac{ax + b}{cx + d}\) (degree of numerator = degree of denominator), the horizontal asymptote is \( y=\frac{a}{c}\). Here, \( a = 1\), \( c = 1\), so \( y = 1\).

Step3: Find x - intercept

Set \( f(x)=0\): \(\frac{x + 3}{x - 1}=0\). A fraction is zero when numerator is zero (denominator non - zero). So \( x+3 = 0\) gives \( x=-3\).

Step4: Find y - intercept

Set \( x = 0\) in \( f(x)\): \( f(0)=\frac{4}{0 - 1}+1=-4 + 1=-3\). So the y - intercept is at \( y=-3\).

Answer:

• Vertical Asymptote: \( x = 1\)
• Horizontal Asymptote: \( y = 1\)
• x - intercept: \( x=-3\) (point \((-3,0)\))
• y - intercept: \( y=-3\) (point \((0, - 3)\))