Figure 1.
First, enter the equation for the given function, f(x)=sqrt((12*x^3 - 5*x + 2)/(1 + 4*x^2 + 3*x^3)), as seen in Figure 1.
Figure 1.
Press the Enter key and the graph of f will appear, as seen in Figure 2.
Figure 2.
It would appear that as we move to the right, the graph in Figure 2 seems to level out at y=2. We can do a calculation to determine if this is the case. You can download the calculation at the link that follows.
In Geogebra, let's enter the equation y=2, as shown in Figure 3.
Figure 3.
Press the Enter key to draw the graph of y=2. The result is shown in Figure 4.
Figure 4.
Note that the graph of f appears to approach the line y=2 asymptotically as x approaches positive infinity.
Open Questions: Can you explain what happens to the graph of f near x = -1? Can you find the equation of the vertical asymptote in the image in Figure 4?