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.
$f(x) = \frac{5}{x + 5} + 5$
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

Vertical asymptote occurs where the denominator is zero (and numerator non - zero). For \(f(x)=\frac{5}{x + 5}+5=\frac{5+5(x + 5)}{x + 5}=\frac{5+5x+25}{x + 5}=\frac{5x + 30}{x + 5}\) (after combining terms), the denominator \(x+5 = 0\) when \(x=-5\). So vertical asymptote is \(x=-5\).

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, when we simplify \(f(x)=\frac{5x + 30}{x + 5}\) (the degree of numerator and denominator is 1). The leading coefficients are 5 (numerator) and 1 (denominator). So horizontal asymptote is \(y = 5\).

Step3: Find x - intercept

Set \(y = 0\), so \(0=\frac{5x+30}{x + 5}\). Cross - multiply (note \(x
eq - 5\)): \(5x+30=0\Rightarrow5x=-30\Rightarrow x=-6\). So x - intercept is \((-6,0)\).

Step4: Find y - intercept

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

Answer:

Vertical Asymptote: \(x=-5\)
Horizontal Asymptote: \(y = 5\)
x - intercept: \((-6,0)\)
y - intercept: \((0,6)\) (To plot: Drag the vertical asymptote to \(x=-5\), horizontal asymptote to \(y = 5\), x - intercept to \((-6,0)\) and y - intercept to \((0,6)\) on the graph)