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) = (3x - 12)/(x² - 16) plot rational function vertical asymptote horizontal asymptote x-intercept y-intercept hole click on a feature then drag it into place. graph with x-axis from -10 to 10 and y-axis from -6 to 10

Explanation:

Step1: Simplify the function

Factor numerator and denominator:
Numerator: \( 3x - 12 = 3(x - 4) \)
Denominator: \( x^2 - 16 = (x - 4)(x + 4) \)
Simplify (cancel \( x - 4 \), \( x
eq 4 \)): \( f(x) = \frac{3}{x + 4} \) (with a hole at \( x = 4 \)).

Step2: Find Vertical Asymptote

Set denominator (of simplified function) to zero: \( x + 4 = 0 \Rightarrow x = -4 \).

Step3: Find Horizontal Asymptote

Degree of numerator (1) < degree of denominator (2) in original function? Wait, simplified is \( \frac{3}{x + 4} \), degree of numerator (0) < degree of denominator (1), so horizontal asymptote \( y = 0 \).

Step4: Find x-Intercept

Set numerator (simplified) to zero: \( 3 = 0 \) (no solution), so no x-intercept. Wait, original numerator: \( 3(x - 4) \), set to zero: \( x = 4 \), but that's a hole, so no x-intercept.

Step5: Find y-Intercept

Set \( x = 0 \) in simplified function: \( f(0) = \frac{3}{0 + 4} = \frac{3}{4} = 0.75 \).

Step6: Find Hole

Hole at \( x = 4 \) (where \( x - 4 \) canceled). Plug \( x = 4 \) into simplified function: \( f(4) = \frac{3}{4 + 4} = \frac{3}{8} = 0.375 \). So hole at \( (4, \frac{3}{8}) \).

Answer:

• Vertical Asymptote: \( x = -4 \) (plot at \( x = -4 \))
• Horizontal Asymptote: \( y = 0 \) (plot along \( y = 0 \))
• y-Intercept: \( (0, \frac{3}{4}) \) (plot at \( (0, 0.75) \))
• Hole: \( (4, \frac{3}{8}) \) (plot at \( (4, 0.375) \))
• x-Intercept: None (no feature to plot)