Sketch the graph of the rational function f(x) = 2x2 − x / x2 − 16 showing all the key features of the graph. label the key
features on your graph.

When x ⇒ 4⁺  (from the rigth), f(x) ⇒+∞

When x ⇒ 4⁻  (from the left), f(x) ⇒-∞

When x ⇒ -4⁺, f(x) ⇒-∞

When x ⇒ -4⁻, f(x) ⇒+∞

Asymptotes:

x = 4

x = -4

y = 2

Step-by-step explanation:

Hi there!

Let´s write the function:

f(x) = (2x² - x) / (x² - 16)

Let´s find the x-value at which the denominator is zero:

x² - 16 = 0

x² = 16

√x² =√16

x = 4 and x = -4

Let´s evaluate the behavior of the function when x ⇒ 4 and x⇒-4 to see if x = 4 and x = -4 are vertical asymptotes:

When x ⇒ 4⁺  (from the rigth)

f(x) ⇒  28/ 0 ⇒ +∞  

when x ⇒ 4⁻ (from the left)

f(x) ⇒ 28 / 0 ⇒ -∞ because x² -16 <0 for being x²< 16

When x ⇒ -4⁺

f(x) ⇒ 36/0 ⇒ -∞ (for the same reason as when x ⇒ 4⁻)

when x ⇒ -4⁻

f(x) ⇒ 36/0 ⇒ +∞

Then, x = 4 and x = -4 are vertical asymptotes.

Let´s evaluate the end behaivor of the function:

f(x) = (2x² - x) / (x² - 16)

When x ⇒ +∞

f(x) ⇒ (2x² - x) / x²

f(x) ⇒ 2x²/x² - x/x²

f(x) ⇒ 2 - 1/x  (1/x ⇒ 0 because 1 divided by a very big number is nearly zero). Then:

f(x) ⇒ 2

When x ⇒ -∞

f(x) ⇒ 2 - 1/x ⇒ 2

Then, y = 2 is a horizontal asymptote.

both equations and proportions are better.