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.
$f(x) = \frac{-x - 3}{2x - 7}$

Explanation:

Step1: Find Vertical Asymptote

Set denominator equal to zero: $2x - 7 = 0$. Solve for $x$: $2x = 7$ $\Rightarrow$ $x = \frac{7}{2} = 3.5$. So vertical asymptote is $x = 3.5$.

Step2: Find Horizontal Asymptote

Compare degrees of numerator and denominator (both degree 1). The ratio of leading coefficients: $\frac{-1}{2}$. So horizontal asymptote is $y = -\frac{1}{2} = -0.5$.

Step3: Find x - intercept

Set numerator equal to zero: $-x - 3 = 0$ $\Rightarrow$ $-x = 3$ $\Rightarrow$ $x = -3$. So x - intercept is $(-3, 0)$.

Step4: Find y - intercept

Set $x = 0$ in $f(x)$: $f(0) = \frac{-0 - 3}{2(0) - 7} = \frac{-3}{-7} = \frac{3}{7} \approx 0.43$. So y - intercept is $(0, \frac{3}{7})$.

Step5: Test for Symmetry (Optional, but to check)

Check if $f(-x) = -f(x)$ (odd) or $f(-x) = f(x)$ (even). $f(-x) = \frac{-(-x) - 3}{2(-x) - 7} = \frac{x - 3}{-2x - 7}$, which is not equal to $f(x)$ or $-f(x)$. So no symmetry.

Answer:

• Vertical Asymptote: $x = 3.5$
• Horizontal Asymptote: $y = -0.5$
• x - intercept: $(-3, 0)$
• y - intercept: $(0, \frac{3}{7})$ (or approximately $(0, 0.43)$)
• No symmetry (additional check)