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.
$f(x) = \frac{6x - 27}{16x + 72}$

Explanation:

Step1: Find Vertical Asymptote

Set denominator equal to zero: $16x + 72 = 0$. Solve for $x$: $16x = -72 \implies x = \frac{-72}{16} = -\frac{9}{2} = -4.5$.

Step2: Find Horizontal Asymptote

For rational function $\frac{ax + b}{cx + d}$, horizontal asymptote is $y = \frac{a}{c}$. Here, $a = 6$, $c = 16$, so $y = \frac{6}{16} = \frac{3}{8} = 0.375$.

Step3: Find x - intercept

Set numerator equal to zero: $6x - 27 = 0 \implies 6x = 27 \implies x = \frac{27}{6} = \frac{9}{2} = 4.5$. So x - intercept is $(4.5, 0)$.

Step4: Find y - intercept

Set $x = 0$ in $f(x)$: $f(0) = \frac{6(0)-27}{16(0)+72} = \frac{-27}{72} = -\frac{3}{8} = -0.375$. So y - intercept is $(0, -0.375)$.

Step5: Check for Holes

Since numerator and denominator have no common factors (GCD of 6x - 27 and 16x + 72 is 1), there are no holes.

Answer:

• Vertical Asymptote: $x = -4.5$
• Horizontal Asymptote: $y = 0.375$
• x - intercept: $(4.5, 0)$
• y - intercept: $(0, -0.375)$
• No Holes