Question

Plot a graph of the rational funcron f(x)=1/x²-16 and label the coordinates of the atationary poiner and inilection points Show the horianeal and vertical anymptots and bbel them with their equutions, Label point(v), if any. where the graph crosser a horizontal aymphise, Cbock yourwork widia graphing uetlity. Enter the tallowing infermution from your gaph (for malnple answers entre exch spparaed by commas [e.g (a)0,2 or (c) (−2,3), (0,4)} if no value enter "none" (a) Verrical Arymptotes z = (b) Horizontal Awmiptores y (6) Poines where the groph crosses i lorisoneal asyuptone (x,y)= (a) featinnary Point (z,y)= (c) Itrifection ivoint (z,y)=

Answer 1

To graph the rational function f(x) = 1/(x²-16), we need to find the vertical asymptotes, horizontal asymptotes, stationary points, and inflection points. The vertical asymptotes are x = -4 and x = 4, and the horizontal asymptote is y = 0. The function has a stationary point at (0, f(0)).

Step-by-step explanation:

To graph the rational function f(x) = 1/(x²-16), we need to find the vertical asymptotes, horizontal asymptotes, stationary points, and inflection points.

First, let's find the vertical asymptotes. The denominator of the function, x²-16, becomes zero when x = ±4. Therefore, the vertical asymptotes are x = -4 and x = 4.

Next, let's find the horizontal asymptotes. We need to consider the behavior of the function as x approaches positive infinity and negative infinity. As x gets larger and larger, the function approaches 0. Therefore, the function has a horizontal asymptote at y = 0.

To find the stationary points and inflection points, we need to differentiate the function f(x) and find where its derivative is zero. The derivative of f(x) is f'(x) = (-2x)/(x²-16)². Setting f'(x) = 0, we get -2x = 0, which gives us x = 0. Therefore, the function has a stationary point at (0, f(0)). To find the inflection points, we need to find the points where the second derivative f''(x) changes sign.

We can display all this information on the graph and label the relevant points.