Question

In Exercises \(57-74\), sketch the graph of the equation. Look for extrema, intercepts, symmetry, and asymptotes as necessary. Use a graphing utility to verify your result. $$ y=\frac{2 x}{1-x^{2}} $$

Short answer

The graph of the function \(y=\frac{2x}{1-x^{2}}\) intersects the origin, shows no particular symmetry, has vertical asymptotes at \(x=-1, 1\) and a horizontal asymptote at \(y=0\). It is positive for \(x>-1\) and \(x<1\), and negative for \(x<-1\) and \(x>1\).

Step-by-step solution

Identify the x and y-intercepts

To find the x-intercepts, set \(y=0\), solve for \(x\). This occurs when \(x=0\). To find the y-intercept, enter \(x=0\) into the equation, finding \(y=0\). Thus, the graph intercepts the origin (0,0).

Determine the Symmetry

The function is neither even nor odd because it doesn’t satisfy the conditions \(f(-x) = f(x)\) for even functions or \(f(-x) = -f(x)\) for odd functions.

Identify the asymptotes

Equating the denominator to zero gives \(x^{2}=1\). Solve for x obtaining \(x=-1, 1\). These are vertical asymptotes. As the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator, the function will have a horizontal asymptote at \(y=0\)

Sketch the graph

The graph intercepts origin, it is neither even nor odd, and has vertical asymptotes at \(x=-1, 1\), horizontal asymptote at \(y=0\). Plot these on your graphing calculator or utility, including the points and asymptotes. Note that for \(x<1\) and \(x>-1\), the function is positive and for \(x<-1\) and \(x>1\), the function is negative.

Verify your result

Use a computer algebra system or a graphing calculator to graph the function, verifying your sketch and attributes.