How do you graph #f(x)=(x-2) / (x+2)#?

Question

Eli Assouline · Accepted Answer

Just to run through some general points:

#f(x) = (x-2) / (x+2)#
What is f(x) when x = 0?
#(0-2)/(0+2) = -2/2 = -1#
Thus our first point on the graph is (0,-1).
What is x at 1?
#(1-2)/(1+2) = -1/3#
Thus our second point on the graph is (1, -1/3).
We continue in this manner until you have enough points on the graph.

One thing to note: when #x=-2#, we have #4/0# (we can't divide by zero!) so our graph will not reach x=-2. Thus we can see an asymptote in our graph at x=-2. As we can see on the y axis, when x is positive it never reaches #f(x) = 1# (as we will always be dividing x by the larger number x+2). The opposite is true when x is negative, as the nominator x will always be larger than the denominator until x=0. graph{(x-2)/(x+2) [-10, 10, -5, 5]}

Kennedy Adams · Answer

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

1. Determine the vertical asymptotes by setting the denominator equal to zero and solving for x. In this case, x+2 = 0, so x = -2 is a vertical asymptote.

2. Determine the horizontal asymptote by comparing the degrees of the numerator and denominator. Since both have a degree of 1, the horizontal asymptote is y = 1.

3. Find the x-intercept by setting the numerator equal to zero and solving for x. In this case, x-2 = 0, so x = 2 is the x-intercept.

4. Determine the behavior of the function as x approaches positive and negative infinity. As x approaches positive infinity, f(x) approaches the horizontal asymptote y = 1. As x approaches negative infinity, f(x) also approaches y = 1.

5. Plot the vertical asymptote at x = -2, the horizontal asymptote at y = 1, and the x-intercept at x = 2.

6. Choose additional x-values to evaluate the function and plot corresponding points on the graph.

7. Connect the plotted points smoothly, avoiding the vertical asymptote.

The resulting graph should show a vertical asymptote at x = -2, a horizontal asymptote at y = 1, an x-intercept at x = 2, and the curve of the function passing through the plotted points.