Problem 60 - 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. $$ y=\frac{2 x}{9-x^{2}} $$

Step-by-Step Solution

Answer

The graph of the function \( y=\frac{2x}{9-x^{2}} \) is symmetric with respect to the origin, has intercepts at the origin (0,0), has vertical asymptotes at \( x = 3 \) and \( x = -3 \), has a horizontal asymptote at \( y = 0 \), and increases without bound as it nears the vertical asymptotes, resembling a hyperbola.

Step 1: Identify the Intercepts

The x-intercept happens where the graph crosses the x-axis, meaning \( y = 0 \). Since \( y = 0 \) when \( x = 0 \), the x-intercept is at point (0,0). The y-intercept happens where the graph crosses the y-axis, this is also at point (0,0) because when \( x = 0, y = 0 \).

Step 2: Check for Symmetry

The graph of the function is symmetric about the origin if replacing \( x \) with \( -x \) yields the same function. The original function is \( y=\frac{2x}{9-x^{2}} \), replacing \( x \) with \( -x \) gives \( y=\frac{-2x}{9-(-x)^{2}} = -y \). This indicates that the function is symmetric with respect to the origin.

Step 3: Find the Asymptotes

Vertical asymptotes occur when the denominator of a fraction is zero. Set \( 9 - x^{2} = 0 \) to solve for \( x \). This happens when \( x = 3 \) and \( x = -3 \), so these are the vertical asymptotes. A horizontal asymptote is where the function heads for as \( x \) goes to \( \pm \) infinity. Using the rule that when the degree of the denominator is greater than the degree of the numerator, the x-axis (y = 0) is the horizontal asymptote.

Step 4: Identify Extrema

The extrema for a function occur at the critical points, where the first derivative equals zero or does not exist. For the function \( y=\frac{2x}{9-x^{2}} \), the derivative \( y' = \frac{18}{(9-x^{2})^{2}} \) does not equal zero anywhere but is undefined at \( x = 3 \) and \( x = -3 \), the same as the asymptotes. These are not extrema, but discontinuities.

Step 5: Create the Graph

Now with all this information, create a graph. Mark the intercept at the origin, the vertical asymptotes at \( x = 3 \) and \( x = -3 \), and the horizontal asymptote at \( y = 0 \). Draw the function, making sure to curve toward the asymptotes as needed. The graph should look like a hyperbola, increasing without bound as it approaches the vertical asymptotes, and symmetric about the origin because of the origin symmetry.

Key Concepts

Vertical Asymptotes

Vertical asymptotes appear in rational functions where the denominator is equal to zero, leading the function to have a division by zero. This creates a point of discontinuity, where the graph cannot reach or cross certain lines. For the function \[ y = \frac{2x}{9-x^2} \], to find the vertical asymptotes:

• Set the denominator to zero: \[ 9 - x^2 = 0 \].
• Solve for \( x \): \[ x^2 = 9 \].
• Take the square root of both sides to find: \( x = 3 \) and \( x = -3 \).

Hence, vertical asymptotes are at \( x = 3 \) and \( x = -3 \). As the function approaches these values, it tends to infinity or negative infinity. It's important to remember: the graph will never actually touch or cross these vertical lines. These are boundaries of infinite or undefined behavior within the function.

Symmetry in Graphs

Symmetry in graphs can provide valuable insights into the behavior of a function. Symmetry makes it easier to understand and draw the graph, as recognizing a symmetrical pattern means you can replicate one side onto the other.

To check for symmetry about the origin, replace \( x \) with \( -x \) in the function and observe if the function becomes its negative. In our function \[ y = \frac{2x}{9-x^2} \], substituting \( -x \) leads to \[ y = \frac{-2x}{9-x^2} = -y \].

• This confirms the function is symmetric with respect to the origin.
• Origin symmetry means that if \( (a, b) \) is on the graph, then \( (-a, -b) \) will also be on the graph.

This type of symmetry often results in graphs that have rotational symmetry around the origin, looking the same upside-down and right-side-up.

Intercepts

Intercepts are the points where a graph crosses either the x-axis or the y-axis. These points are crucial for sketching the graph accurately.

**Finding the x-intercept:** The x-intercept occurs where \( y = 0 \). For our function \[ y = \frac{2x}{9-x^2} \]:

• Set \( y = 0 \), which happens when the numerator is zero.
• This gives an x-intercept at \( x = 0 \), or the point \((0, 0)\).

**Finding the y-intercept:** The y-intercept happens where \( x = 0 \):

• Insert \( x = 0 \) into the function, \( y = \frac{2(0)}{9-(0)^2} = 0 \).
• This shows the y-intercept is also at \((0, 0)\).

Thus, for this specific function, both intercepts coincide at the origin, \((0, 0)\), providing a central point from which the symmetry in the graph emerges.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as \( x \) approaches positive or negative infinity. They help in predicting the end behavior of a graph's curve.

In the function \[ y = \frac{2x}{9-x^2} \], we analyze the degrees of the polynomials in the numerator and the denominator to determine horizontal asymptotes.

• If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is at \( y = 0 \).
• Here, the degree of the numerator (1) is less than the degree of the denominator (2).
• This confirms that as \( x \to \pm \infty \), the graph approaches the line \( y = 0 \).

The graph gets closer and closer to the x-axis but never actually touches it. This horizontal asymptote illustrates how the values of \( y \) diminish towards zero as \( x \) moves infinitely away from the origin.