Graph the rational function

Question

Graph the rational function.

f(x)= -x^2+4x-1/ x-2

Start by drawing the asymptotes. Then plot two points on each piece of the graph

Answer 1

Starting with the easiest part, the vertical asymptote is the value of x that would cause the denominator to be zero (making the function undefined at that point). If x-2 = 0, then x = 2. That's the vertical asymptote.

Now we'll look for another asymptote. Horizontal asymptotes exist if the degrees of the numerator and denominator are the same or if the degree of the numerator is smaller. That's not the case here. The degree of the numerator (2) is higher; therefore we have a slant asymptote. To find its formula, perform the division defined by the rational expression.

(-x²+4x-1) / x-2

This gives us -x+2 and a remainder. The remainder is ignored, so the formula for the slant asymptote is -x+2

Draw these asymptotes.

At this point, you should have some idea of what the graph will look like. If not, you can experiment with which x values to input in order to plot points that will be useful in sketching the graph. Remember, you want to select values that are fairly close to the bend in the graph.

In this case, for the lower graph, try inputting -1, 0, and 1

For the upper curve, try x = 3, 4, and 5 (note that we must avoid 2 but still be kinda close to it)

Plug those values in, keeping in mind that the graphs aren't allowed to cross the asymptotes, and you're all set.

Best wishes!