Problem 16 - Algebra and Trigonometry

Question

Determine an appropriate viewing rectangle for the equation and use it to draw the graph. $$ y=\frac{x}{x^{2}+25} $$

Step-by-Step Solution

Answer

Use viewing rectangle \([-10, 10]\) for x-axis and \([-0.5, 0.5]\) for y-axis.

Step 1: Understand the Function Behavior

First, analyze the function \( y = \frac{x}{x^2 + 25} \). As \( x \) increases, the denominator \( x^2 + 25 \) grows much faster than the numerator \( x \), suggesting that the function will tend towards 0 asymptotically as \( x \to \pm\infty \). At \( x = 0 \), the function simplifies to \( y = 0 \). Thus, the function is odd and symmetric about the origin.

Step 2: Identify Intercepts

Find any intercepts by setting \( y = 0 \). Solving the equation \( \frac{x}{x^2 + 25} = 0 \) results in \( x = 0 \). This indicates a single intercept at the origin (0,0). There are no additional x-intercepts or y-intercepts.

Step 3: Analyze Function Symmetry

Verify symmetry about the origin. The function \( y = \frac{x}{x^2 + 25} \) is odd, meaning \( f(-x) = -f(x) \). When \( x \) is replaced with \( -x \), we have \( f(-x) = \frac{-x}{x^2 + 25} = -\frac{x}{x^2 + 25} = -f(x) \). Thus, the function is symmetric about the origin.

Step 4: Determine Asymptotic Behavior

As \( x \to \infty \) or \( x \to -\infty \), \( y = \frac{x}{x^2 + 25} \to 0 \). The horizontal asymptote is at \( y = 0 \); the graph will approach this line but never touch or cross it.

Step 5: Choose Initial Viewing Rectangle

Considering the intercept, symmetry, and asymptotic behavior, a viewing rectangle roughly symmetric about the origin should suffice. Choose \([-10, 10]\) for the x-axis, ensuring the behavior near zero and as \( x \) grows large can be observed. For the y-axis, \([-0.5, 0.5]\) will show the approach to the horizontal asymptote.

Key Concepts

Function Symmetry

In graphing rational functions, understanding symmetry is a vital step. Symmetry in a function can simplify the graphing process by revealing repetitive behavior. A function is symmetric about the y-axis if replacing \( x \) with \( -x \) produces the same function. Such functions are called even. Conversely, a function is odd if replacing \( x \) with \( -x \) produces the negative of the original function, meaning \( f(-x) = -f(x) \). In our given function \( y = \frac{x}{x^2 + 25} \), symmetry is demonstrated by checking the equation \( f(-x) = -f(x) \).

• When you substitute \( x \) with \( -x \), the function becomes \( \frac{-x}{x^2 + 25} \), which simplifies to \(-\frac{x}{x^2 + 25} \).
• This confirms that the function is odd and symmetric about the origin. This inherent symmetry helps in predicting the behavior and sketching the graph.

Using this property, you only need to study half of the graph to understand the entire function's behavior.

Asymptotic Behavior

Asymptotic behavior is another crucial concept in graphing rational functions. An asymptote is a line that a graph approaches but never actually touches. For the function \( y = \frac{x}{x^2 + 25} \), as \( x \) increases or decreases without bound, the denominator \( x^2 + 25 \) grows significantly larger than the numerator \( x \). This causes the fraction to become closer and closer to zero, indicating a horizontal asymptote at \( y = 0 \).

• The graph approaches this asymptote infinitely, getting closer but never intersecting.
• This behavior defines how the tails of the graph act at the extremes of the x-axis.

Understanding asymptotic behavior helps in choosing the right scale for the y-axis in your graphing window, ensuring you capture the essence of the function's behavior as \( x \to \pm\infty \).

Viewing Rectangle

Choosing an appropriate viewing rectangle is essential for accurately representing the behavior of rational functions. A viewing rectangle is a range of x and y-values that allows observation of the function's significant features. For \( y = \frac{x}{x^2 + 25} \):

• The intercept at the origin, symmetry about the origin, and horizontal asymptote suggest focusing on a symmetric window around the origin.
• An x-range of \([-10, 10]\) is recommended to clearly observe behavior as \( x \) changes.
• For the y-axis, a range of \([-0.5, 0.5]\) effectively captures the approach towards the asymptote \( y = 0 \).

By selecting these dimensions, you can ensure that all key behavior, including the trend towards the asymptote and origin intercept, is visible.

Intercepts of Rational Functions

Finding the intercepts of rational functions helps identify key intersections with the axes. Intercepts are where the graph crosses the x-axis and y-axis, providing anchor points for sketching the function. To find x-intercepts, set \( y = 0 \) and solve the equation. In \( y = \frac{x}{x^2 + 25} \):

• The numerator \( x \) must equal 0, resulting in an x-intercept at \( x = 0 \), which is the origin (0,0).
• For y-intercepts, set \( x = 0 \) and solve for \( y \). Here, this step leads to \( y = 0 \), confirming no additional y-intercepts.
• This shows that the rational function intersects the x-axis at the origin, and there are no further intercepts.

By identifying these intercepts, we have precise points to plot and a baseline for sketching additional function behavior.