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. $$ y=\frac{2 x}{1-x} $$

Short answer

The graph of \(y=\frac{2x}{1-x}\) has asymptotes at \(x=1\) (vertical) and \(y=0\) (horizontal). The x and y intercepts are both 0 and there's no symmetry. The curve approaches the asymptotes without touching them.

Step-by-step solution

Determine the vertical asymptote

A vertical asymptote occurs when the denominator of a fraction is equal to zero. Therefore, set \(1-x=0\), and solve for \(x\). Thus, \(x = 1\) is the vertical asymptote.

Determine the horizontal asymptote

A horizontal asymptote for a rational function can be found by comparing the degrees of the top and bottom polynomials. The horizontal asymptote in this case would be \(y=0\) because the degree of the denominator is greater than the degree of the numerator.

Determine the x-intercepts

An x-intercept is where the graph crosses the x-axis. This occurs when \(y=0\). Set the equation \(y=\frac{2x}{1-x}\) equal to zero and solve for \(x\). The equation will be 0 if the numerator \(2x = 0\), thus \(x = 0\) is the x-intercept.

Determine the y-intercepts

A y-intercept is where the graph crosses the y-axis. This occurs when \(x=0\). Substitute \(x=0\) into the function equation \(y=\frac{2x}{1-x}\), and the result will be \(y = 0\). So, the y-intercept is also at 0.

Check for symmetry

To check for symmetry replace \(x\) with \(-x\) in the equation and simplify. If the result is identical to the original function, the function is symmetric about the y-axis. If it's the negative of the original function, then it's symmetric about the origin. Substituting \(-x\) in place of \(x\) in the equation gives \(y=\frac{-2x}{1+x}\), which isn't identical or negative of the original equation. Therefore, the function isn't symmetric.

Sketch the graph

Firstly, sketch the vertical line \(x=1\) to indicate the vertical asymptote. Then, sketch the horizontal line \(y=0\) to define the horizontal asymptote. Next, mark the x and y intercepts which are both at 0. Because there's no symmetry, and since \(x < 1\), \(y\) will tend towards negative infinity and as \(x > 1\), \(y\) will tend towards positive infinity. Sketch the curve following these conditions.

Verify with a graphing utility

Use a graphing calculator or an online graphing tool to draw the equation \(y=\frac{2x}{1-x}\). The graph should confirm the manually sketched graph.