Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-30, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\)-intercepts. $$ y=\frac{x}{x^{2}+1} $$

Step-by-Step Solution

Answer

The graph is symmetric about the origin, intercepts at \((0,0)\), and has a horizontal asymptote at \(y=0\).

Step 1: Identify Symmetries

To determine symmetries, check for symmetry about the y-axis, x-axis, and origin. Replace \(x\) with \(-x\) in the equation to check for y-axis symmetry. The equation becomes \(y = \frac{-x}{x^2 + 1}\), which is not equal to the original, indicating no y-axis symmetry. For origin symmetry, replace \(x\) with \(-x\) and \(y\) with \(-y\), giving \(-y = \frac{-x}{x^2 + 1}\), which simplifies to the original equation: \(y = \frac{x}{x^2 + 1}\). Therefore, the graph has origin symmetry.

Step 2: Find x-Intercepts

To find the x-intercepts, set \(y = 0\) and solve for \(x\). The equation \(0 = \frac{x}{x^2 + 1}\) implies \(x = 0\). Therefore, the x-intercept is at \((0, 0)\).

Step 3: Find y-Intercepts

To find the y-intercepts, set \(x = 0\) and solve for \(y\). Substituting, we get \(y = \frac{0}{0^2 + 1} = 0\). Therefore, the y-intercept is at \((0, 0)\).

Step 4: Analyze Asymptotic Behavior

Examine the behavior of \(y\) as \(x\) approaches infinity or negative infinity. For large \(|x|\), \(y = \frac{x}{x^2 + 1}\) approaches \(y \approx \frac{x}{x^2} = \frac{1}{x}\), indicating that \(y\) approaches 0 but never actually reaches it. Thus, the equation has a horizontal asymptote at \(y = 0\).

Step 5: Sketch the Graph

Plot the points found: the intercept at \((0, 0)\), and the asymptotic behavior as \(x\) grows large in either direction. The function is odd and symmetric about the origin. As \(x\) moves away from zero, the value of \(y\) approaches but never reaches 0. The curve will reside primarily in the first and third quadrants, moving from the second to the fourth as \(x\) crosses zero.

Key Concepts

Symmetry in Graphs

Symmetry in a graph helps us understand the overall shape and behavior of a function. For most graphs, we typically look for symmetry in three ways: around the y-axis, the x-axis, and the origin.
To check for y-axis symmetry, replace every instance of \(x\) in the equation with \(-x\). For our equation, this results in \(y = \frac{-x}{x^2 + 1}\). Since this does not match the original equation, there is no y-axis symmetry. Next, for origin symmetry, substitute \(x\) with \(-x\) and \(y\) with \(-y\), leading to \(-y = \frac{-x}{x^2 + 1}\). Simplifying this, we find that it results in the original equation, confirming origin symmetry.
This origin symmetry indicates that the graph is a mirror image of itself when rotated 180 degrees around the origin. Such symmetry is often seen in odd functions.

X-Intercepts and Y-Intercepts

Intercepts are points where the graph crosses the axes. They provide key information about where the function is zero (for x-intercepts) and the starting value of the graph (for y-intercepts).
For x-intercepts, we set \(y = 0\) and solve for \(x\). In the equation \(0 = \frac{x}{x^2 + 1}\), multiplying both sides by \(x^2 + 1\) gives \(x = 0\). Thus, the x-intercept is at the origin, or point \((0,0)\).
For y-intercepts, set \(x = 0\) and solve for \(y\). Plugging into the equation gives \(y = \frac{0}{0^2 + 1} = 0\), confirming the y-intercept is also at \((0,0)\).
The location of both intercepts at \((0,0)\) gives further insights into the steepness and orientation of the graph at the origin.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as it approaches infinity or an undefined point. In simpler terms, it's about tracking where the function is headed without ever actually reaching there.
For the function \(y = \frac{x}{x^2 + 1}\), as \(x\) becomes very large (positive or negative), the term \(x^2 + 1\) dominates, and \(y\) approximates to \(\frac{x}{x^2} = \frac{1}{x}\).
As \(x\) approaches infinity, \(\frac{1}{x}\) approaches 0, signaling a horizontal asymptote at \(y = 0\).

• This means that, no matter how large or small \(x\) gets, the value of \(y\) gets closer to zero without ever touching it.

This behavior affects the overall shape of the graph and is important in sketching an accurate plot.

Sketching Graphs

Sketching graphs combines the information of intercepts, symmetry, and asymptotic behavior to visualize a function's behavior. With all these insights, you can better predict the graph's shape.
Start by marking the intercepts, in this case at \((0,0)\). Knowing the graph's origin symmetry, you can expect it to mirror along the origin.

• The asymptotic behavior showing \(y \rightarrow 0\) as \(|x|\) grows indicates the graph will flatten out near the x-axis.
• The symmetry about the origin tells us the graph appears in the opposite quadrant of any plotted part.

Finally, draw a smooth curve through the intercepts and adjust it to approach but not touch the x-axis, flowing through the first and third quadrants based on the direction from the original symmetry. This comprehensive approach ensures an accurate and meaningful representation of the function.