Problem 68 - Essential Calculus

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

Step-by-Step Solution

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 1: 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).

Step 2: 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.

Step 3: 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 \(y=0\)

Step 4: 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.

Step 5: Verify your result

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

Key Concepts

Understanding Functions

Functions are mathematical entities that relate an input to an output. In this specific exercise, the function given is \( y = \frac{2x}{1-x^2} \). The role of a function is to assign exactly one output for each input from the domain. When analyzing functions like this, it helps to identify certain key characteristics such as intercepts, asymptotes, and symmetry. In this exercise, understanding how each part of the function behaves helps sketch a more accurate graph.

The Role of Asymptotes

Asymptotes are lines that a graph approaches but never touches. For the function \( y = \frac{2x}{1-x^2} \), there are vertical asymptotes at \( x = -1 \) and \( x = 1 \). This is because the denominator of the function becomes zero at these points, causing the function to break or become undefined.

• Vertical asymptotes happen when the denominator is zero, and they represent values that the function will never reach.
• Horizontal asymptotes, such as \( y = 0 \) for this function, occur when the degree (or highest power) of the polynomial in the numerator is less than that in the denominator.

Asymptotes are crucial in graph sketching as they help define the boundaries of the function's behavior.

Determining Intercepts

Intercepts are the points where the graph crosses the axes. The x-intercepts are found by setting the entire function equal to zero (\( y = 0 \)) and solving for \( x \). For this function, \( x = 0 \) is the x-intercept. Similarly, the y-intercept is found by setting \( x = 0 \) and solving for \( y \), which also gives \( y = 0 \). This means the graph passes through the origin (0,0).
Intercepts provide significant information about where the graph exists in relation to the axes, offering anchor points that ground the structure of the graph.

Analyzing Symmetry

Symmetry in functions refers to whether a graph looks the same when flipped or rotated across an axis or the origin. For even symmetry, a function satisfies \( f(-x) = f(x) \), and for odd symmetry, it satisfies \( f(-x) = -f(x) \). In this exercise, the function does not satisfy either condition, indicating it has no symmetry relative to the axes.
Understanding symmetry helps simplify graphing processes because it reveals how parts of the graph duplicate or mirror each other. Lack of symmetry means the graph needs to be considered as a whole rather than in mirrored sections.