How do you graph #y =(2x(x-2)) / ((x-3) (x+1))#?

First, find the vertical asymptote. The vertical asymptote is the value of x that will make the denominator equal to 0.

Graph x=3 and x=-1 using a dotted line.

Next, check whether the equation has a horizontal asymptote or a slant asymptote. The first thing you need to do is to find out the degree of the numerator and denomintor. Expand the equation first.

Now that you're done with the asymptotes, you can start plotting some points. You can do this by simply inserting values. Try starting with simple ones such as by setting x to 0.

It will look like this: graph{(2x^2 - 4x) / (x^2 - 2x - 3) [-40, 40, -20, 20]}

To graph the equation y = (2x(x-2)) / ((x-3)(x+1)), we can follow these steps:

1. Determine the domain of the function by identifying any values of x that would make the denominator zero. In this case, x cannot be equal to 3 or -1.

2. Find the x-intercepts by setting y equal to zero and solving for x. This means setting (2x(x-2)) / ((x-3)(x+1)) = 0. The x-intercepts occur when the numerator is equal to zero, so we solve 2x(x-2) = 0. This gives us x = 0 or x = 2.

3. Determine the y-intercept by substituting x = 0 into the equation. This gives us y = (2(0)(0-2)) / ((0-3)(0+1)) = 0.

4. Analyze the behavior of the function as x approaches positive and negative infinity. As x approaches positive or negative infinity, the function approaches zero.

5. Plot the x-intercepts, y-intercept, and any additional points of interest on the graph.

6. Determine the end behavior of the graph by observing the behavior of the function as x approaches the domain limits (3 and -1). This can be done by evaluating the function for values of x slightly greater and slightly smaller than 3 and -1.

7. Connect the points on the graph smoothly, considering the behavior of the function and any asymptotes if applicable.

By following these steps, you can graph the equation y = (2x(x-2)) / ((x-3)(x+1)).