Question

Sketch the graph of the equation. Use intercepts, extrema, and asymptotes as sketching aids. \(y=\frac{2 x}{1-x}\)

Short answer

The x-intercept of the graph is at \(x=0\) and the y-intercept is at \(y=0\). The graph has a vertical asymptote at \(x=1\) and there is no horizontal asymptote. There are no extrema for this graph and it approaches but does not reach the asymptotes.

Step-by-step solution

Find the intercepts

To find the x-intercept, set y equal to 0 and solve for x. This will give the x-coordinate of the intercept. Similarly, to find the y-intercept, set x equal to 0 and solve for y.

Find the extrema

To find the extrema, the first derivative of the function needs to be found and set equal to 0, then solve for x. However, for the given function there are no extrema as the derivative is always positive.

Find the asymptotes

Vertical asymptotes occur when the denominator of the function equals zero. Therefore, set the denominator 1-x equal to 0 and solve for x. This gives the vertical asymptote. The horizontal asymptote can be determined by looking at the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0. If they are the same, the horizontal asymptote is the ratio of the leading coefficients. In this case, there is no horizontal asymptote, as the degree of the numerator is less than the degree of the denominator.

Sketch the graph

First, draw the x and y axes and mark the intercepts. Since there are no extrema, there are no maximum or minimum points to mark. However, draw the vertical asymptote as a dotted line at x=1. Now, sketch the graph of the function. It should approach but never reach the asymptotes. The graph will also pass through the intercepts.

Key concepts

Intercepts

Intercepts are crucial points where a graph meets the axes. They provide the initial anchors for sketching a rational function.
To find the **x-intercept**, you set the function equal to zero, essentially finding out where the graph crosses the x-axis. For the equation given, we set \(y = 0\) and solve \(\frac{2x}{1-x}=0\), which simplifies to \(2x = 0\). Hence, \(x = 0\) is the x-intercept.
For the **y-intercept**, you set \(x = 0\) and solve for y, which shows where the graph meets the y-axis. Substituting x = 0 in the equation gives \(y = \frac{2(0)}{1-0} = 0\). Therefore, the y-intercept is also at (0,0).
Both intercepts in this case are at the origin, giving a clean point from which to start sketching the curve.

Asymptotes

Asymptotes guide us on behavior as a function approaches specific lines but never quite meets them. They are vital for understanding rational functions.
A **vertical asymptote** occurs where the function becomes undefined, usually where the denominator equals zero. For \(y=\frac{2x}{1-x}\), set \(1-x=0\) to find the vertical asymptote, resulting in \(x=1\). This means the graph will approach but not touch or cross the line \(x=1\).
**Horizontal asymptotes** describe the function's behavior as \(x\) approaches \(\infty\) or \(-\infty\). The degrees of the polynomial in the numerator and denominator determine horizontal asymptotes. Here, since the numerator's degree is less than that of the denominator, the function does not have a horizontal asymptote; instead, it will approach the x-axis \(y=0\) as \(x\) becomes very large or very small.Vertical and horizontal asymptotes help predict where the graph will stretch or compress while sketching.

Extrema

Extrema are the high and low points of a graph, indicating the peaks and valleys.
To locate **extrema**, we differentiate the function and search for where the derivative becomes zero, suggesting a potential maximum or minimum point. However, for the function \(y=\frac{2x}{1-x}\), calculating the derivative reveals it does not equate to zero at any point within the domain. The derivative, \(f'(x) = \frac{2}{(1-x)^2}\), is always positive for \(x < 1\) and hence shows the function is always increasing or decreasing and has no turning points.
In simpler terms, this function lacks significant climbs or drops, steadily moving without peaking or dipping. Understanding this characteristic ensures accuracy in graphing, as we don't add any extra features like peaks or troughs.