Let's go through the problem step by step to graph the rational function [tex]\( f(x) = \frac{4x}{x-4} \)[/tex]:
1. There is no x-intercept.
This means that the function [tex]\( f(x) \)[/tex] does not cross the x-axis. This occurs when the numerator of the function is zero, but since the numerator [tex]\( 4x \)[/tex] is zero only when [tex]\( x=0 \)[/tex], which doesn't yield an x-intercept in this case.
2. Find the vertical asymptote(s):
The vertical asymptotes occur where the denominator of the function is zero.
For [tex]\( f(x) = \frac{4x}{x-4} \)[/tex], the denominator is [tex]\( x-4 \)[/tex]. Setting the denominator equal to zero gives:
[tex]\[
x - 4 = 0 \implies x = 4
\][/tex]
Thus, the vertical asymptote is:
[tex]\[
\boxed{x = 4}
\][/tex]
3. Find the horizontal asymptote(s):
To determine the horizontal asymptote, compare the degrees of the numerator and the denominator. Here, both the numerator and the denominator have the same degree (1). The horizontal asymptote is found by dividing the leading coefficients of the highest degree terms.
The leading coefficient of the numerator [tex]\( 4x \)[/tex] is 4, and the leading coefficient of the denominator [tex]\( x-4 \)[/tex] is 1. Therefore, the horizontal asymptote is:
[tex]\[
\boxed{y = 4}
\][/tex]
4. Plot points between and beyond each x-intercept and vertical asymptote:
Evaluate the function [tex]\( f(x) = \frac{4x}{x-4} \)[/tex] at several points:
[tex]\[
\begin{aligned}
f(-5) &= \frac{4(-5)}{-5-4} = \frac{-20}{-9} = 2.22 \\
f(-1) &= \frac{4(-1)}{-1-4} = \frac{-4}{-5} = 0.8 \\
f(3) &= \frac{4 \cdot 3}{3-4} = \frac{12}{-1} = -12 \\
f(5) &= \frac{4 \cdot 5}{5-4} = \frac{20}{1} = 20 \\
f(6) &= \frac{4 \cdot 6}{6-4} = \frac{24}{2} = 12 \\
\end{aligned}
\][/tex]
So, the function values at these points are:
[tex]\[
\begin{aligned}
f(-5) &= \boxed{\frac{20}{9}} \approx 2.22 \\
f(-1) &= \boxed{\frac{4}{5}} = 0.8 \\
f(3) &= \boxed{-12} \\
f(5) &= \boxed{20} \\
f(6) &= \boxed{12}
\end{aligned}
\][/tex]
With this information, you can now accurately graph the function [tex]\( f(x) = \frac{4x}{x-4} \)[/tex] using the x-intercepts, vertical asymptote, horizontal asymptote, and the evaluated points.
