Question

Sketch the graph of the equation. Use intercepts, extrema, and asymptotes as sketching aids. \(f(x)=\frac{x^{2}}{x^{2}+9}\)

Short answer

The graph of \(f(x)=\frac{x^{2}}{x^{2}+9}\) has an intercept at the origin (0, 0), a horizontal asymptote at y=1, no vertical asymptotes, and a local minimum at x=0. The graph approaches y=1 as x approaches positive or negative infinity.

Step-by-step solution

Finding the Intercept

To find the x-intercept, set the function equal to 0 and solve for x. Here the function \(f(x)=\frac{x^{2}}{x^{2}+9}\) is equal to 0 only when x=0. So, x-intercept is (0, 0). For the y-intercept, simply put x=0 in the equation to find y. Therefore, y-intercept is also (0,0).

Finding the Asymptotes

Here, since the degree of the numerator is equal to the degree of the denominator, we use the ratio of the coefficients of the highest degree terms to find the horizontal asymptote. Hence, the horizontal asymptote is y=1. For vertical asymptotes, we can search where the denominator equals zero. However, since the equation \(x^{2}+9=0\) has no real roots, the graph has no vertical asymptotes.

Finding the Extrema

We use the first derivative of the function \(f'(x)=0\) to find any local minimums or maximums. First derivative of \(f(x)=\frac{x^{2}}{x^{2}+9}\) is \(f'(x)=\frac{18x}{(x^{2}+9)^{2}}\). Setting the derivative equal to zero gives x=0, which is a local minimum position. However, since there's no local maximum position, the graph will extend towards the asymptote.

Sketching the Graph

Now we can sketch the graph. Draw the axes with the intercept at (0, 0). Draw the horizontal asymptote y=1. At x=0, we have a local minimum point. Sketch the graph approaching the horizontal asymptote to the right and left of x=0.

Key concepts

Finding Intercepts

The concept of finding intercepts in a graph refers to determining the points where the graph crosses the x-axis (the x-intercepts) and y-axis (the y-intercepts). For the function \(f(x)=\frac{x^{2}}{x^{2}+9}\), both intercepts coincide at the origin (0,0).

To find the x-intercept of a rational function, set \(f(x)=0\) and solve for x. Here, the function is never zero unless x is zero, making (0,0) the sole x-intercept. Similarly, the y-intercept occurs where x=0. Substituting zero into the function yields \(f(0)=\frac{0}{0+9}=0\), confirming that the y-intercept is also at (0,0).

This is a crucial first step in graphing a rational function as it provides an anchor point from which we can further analyze and construct the graph.

Identifying Asymptotes

An asymptote is a line that the graph of a function approaches but never actually reaches. Asymptotes can be horizontal, vertical, or slanted. For the given function \(f(x)=\frac{x^{2}}{x^{2}+9}\), we identify the horizontal asymptote by comparing the degrees of the polynomial in the numerator and denominator.

Since they are of the same degree, we look at the ratio of the leading coefficients. Both the numerator and denominator have a leading coefficient of 1, yielding a horizontal asymptote at \(y=1\).

For vertical asymptotes, we examine the points where the denominator is equal to zero. However, in this case, \(x^{2}+9=0\) has no real solutions, indicating there are no vertical asymptotes. Understanding where these invisible boundaries lie is paramount for sketching an accurate graph of a rational function.

Graphing Extrema

Extrema refer to the points on a graph where a function reaches its local maximums or minimums. For the function \(f(x)=\frac{x^{2}}{x^{2}+9}\), we identify extrema using derivative analysis.

The first derivative \(f'(x)=\frac{18x}{(x^{2}+9)^{2}}\) helps determine where any potential extrema are by setting the derivative to zero and solving for x. Here, setting \(f'(x)=0\) leads to the solution x=0. This point corresponds to a local minimum as there are no local maximum points for this function. The graph dips to its lowest point at x=0 before rising and approaching its horizontal asymptote. Finding and noting the extrema are key to understanding the curvature and behavior of a graph, giving visual cues to predict the function's movement beyond the plotted range.

Derivative Analysis

Derivative analysis is a mathematical tool used to determine the slope of a function at any point, which in turn can help find rates of change, extrema, and the concavity of the graph. The derivative of our given function \(f'(x)=\frac{18x}{(x^{2}+9)^{2}}\) provides us with essential information.

When the derivative is positive, the function is increasing, and when it is negative, the function is decreasing. By finding where the derivative equals zero, we can identify critical points which could be potential local extrema, as we have with x=0 in this function. Moreover, the function's slope is constantly changing anywhere the derivative is non-zero, which affects how we sketch the graph between its intercepts and as it approaches its asymptotes.

Employing derivative analysis enables us to understand the meticulous behavior of the function, and it is a powerful component when graphing to gain a complete picture of the function's behavior.