How do you graph #f(x)=-2/(x+3)# using holes, vertical and horizontal asymptotes, x and y intercepts?

Question

Isaiah Anthony · Accepted Answer

The vertical asymptote is #x=-3#
The horizontal asymptote is #y=0#
See the graph below

To calculate the vertical asymptote, we perform #lim_(x->-3^+)f(x)=lim_(x->-3^+)-2/(x+3)=-oo# #lim_(x->-3^-)f(x)=lim_(x->-3^+)-2/(x+3)=+oo# The vertical asymptote is #x=-3# To calculate the horizontal asymptote, we perform #lim_(x->+oo)f(x)=lim_(x->+oo)-2/(x+3)=0^-# #lim_(x->-oo)f(x)=lim_(x->-oo)-2/(x+3)=0^+# The horizontal asymptote is #y=0# The general form of the graph is #color(white)(aaaa)##x##color(white)(aaaaaaa)##-oo##color(white)(aaaaaaaa)##-3##color(white)(aaaaaaa)##+oo# #color(white)(aaaa)##-(x+3)##color(white)(aaaaa)##+##color(white)(aaaaaa)##||##color(white)(aaa)##-# #color(white)(aaaa)##f(x)##color(white)(aaaaaaaaa)##↗##color(white)(aaaaaa)##||##color(white)(aaa)##↗# The intercept with the y-axis is when #x=0# #f(0)=-2/3# So the intercept is #(0,-2/3)# graph{-2/(x+3) [-18.02, 18.03, -9.01, 9.01]}

Eli Assouline · Answer

undefined To graph the function f(x) = -2/(x+3), we can follow these steps:

1. Holes: Check if there are any values of x that make the denominator (x+3) equal to zero. In this case, x cannot be -3, as it would result in division by zero. Therefore, there is a hole at x = -3.

2. Vertical Asymptotes: Determine if there are any vertical asymptotes by finding the values of x that make the denominator equal to zero. In this case, x = -3 is the only value that makes the denominator zero. Thus, there is a vertical asymptote at x = -3.

3. Horizontal Asymptotes: To find the horizontal asymptote, we examine the behavior of the function as x approaches positive or negative infinity. In this case, as x approaches positive or negative infinity, the function approaches zero. Therefore, the horizontal asymptote is y = 0.

4. x-intercept: To find the x-intercept, set y (or f(x)) equal to zero and solve for x. In this case, setting -2/(x+3) equal to zero gives us x = -3. So, the x-intercept is -3.

5. y-intercept: To find the y-intercept, set x equal to zero and solve for y (or f(x)). In this case, setting x = 0 gives us y = -2/3. So, the y-intercept is -2/3.

By considering these steps, you can graph the function f(x) = -2/(x+3) accurately.