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. $$ y=\frac{2 x^{2}}{x^{2}+4} $$

Short answer

The graph of the equation \(y = \frac{2x^{2}}{x^{2}+4}\) is a curve symmetric with respect to the y-axis with the x-intercept and y-intercept at the origin and a horizontal asymptote \(y = 2\). There are no vertical asymptotes.

Step-by-step solution

Find x-intercepts

Set \(y = 0\) in the given equation and solve for \(x\). This results in the equation \(0 = \frac{2x^{2}}{x^{2}+4}\). This implies \(x = 0\), so the x-intercept is \(0\).

Determine the y-intercept

Set \(x = 0\) in the given equation and solve for \(y\). This yields \(y = \frac{2*0^{2}}{0^{2} + 4} = 0\). So, the y-intercept is also \(0\).

Check for symmetry

The function \(y = \frac{2x^{2}}{x^{2}+4}\) is even since replacing \(x\) with \(-x\) does not change the function (which means it's symmetric with respect to the y-axis).

Identify asymptotes

The horizontal asymptote can be found by comparing the degree of the numerator and the denominator. Both are 2, so the horizontal asymptote is the ratio of the leading coefficients, which is \(y = 2/1 = 2\). As there is no value of \(x\) for which the denominator is zero, there are no vertical asymptotes.

Use a graphing utility

The graph can be verified with a graphing calculator or online graphing tool, confirming the intercepts, symmetry, and asymptotes determined analytically. It's a curve going through the origin (0,0), symmetric along the y-axis, reaching towards but never touching the line \(y=2\).

Key concepts

X-Intercepts

Understanding the x-intercepts of a graph is crucial for visualizing how a function behaves in relation to the x-axis. An x-intercept is a point where the graph of the function crosses or touches the x-axis, which indicates that the function has a value of zero at that particular point. In our exercise focusing on the function \(y = \frac{2x^2}{x^2+4}\), finding the x-intercepts is achieved by setting \(y\) to zero and solving for \(x\). This gives us the equation \(0 = \frac{2x^{2}}{x^{2}+4}\). After simplifying, we see that the only solution that satisfies this condition is \(x = 0\), giving us an x-intercept at the origin, \((0,0)\).

It's beneficial for students to remember that regarding rational functions, x-intercepts can only occur where the numerator equals zero, while the denominator is not zero to avoid undefined values (division by zero is not allowed).

Y-Intercepts

Similar to x-intercepts, the y-intercept of a graph manifests where the function crosses the y-axis. This occurs when \(x = 0\), reflecting the output value of the function when it's input is zero. For the given function \(y = \frac{2x^2}{x^2+4}\), we determine the y-intercept by setting \(x\) to zero. Plugging in the values, we find that \(y = \frac{2*0^{2}}{0^{2} + 4} = 0\), which again places the y-intercept at the origin, \((0,0)\).

Identifying the y-intercept can be especially helpful as a reference point when graphing a function manually. It serves as a starting point and can assist in understanding the initial behavior of the function.

Symmetry in Functions

The concept of symmetry is a fascinating aspect of graphing functions, where symmetry describes mirror-like behavior of a graph. If a function is even, it means the graph is symmetrical about the y-axis. To check for evenness, we substitute \(x\) with \(-x\) and look for the original function. For our function \(y = \frac{2x^2}{x^2+4}\), replacing \(x\) with \(-x\) yields \(y = \frac{2(-x)^2}{(-x)^2+4}\), which simplifies to the same original function, confirming it is symmetrical about the y-axis.

This symmetry informs us that for every point \((x, y)\) on the graph, there is a corresponding point \((-x, y)\), making plotting more efficient and helping to predict the behavior of the function.

Horizontal Asymptotes

In the realm of rational functions, a horizontal asymptote is a horizontal line that the graph of the function approaches as \(x\) moves towards positive or negative infinity. It represents the limiting behavior of the function. The asymptote can be predicted by comparing the highest power of \(x\) in both the numerator and the denominator. If the degrees are equal, the horizontal asymptote will be the ratio of the leading coefficients. For our example, the degrees are the same, with each being 2, and the leading coefficient in the numerator is 2 while it is 1 in the denominator. Hence, the horizontal asymptote is \(y = \frac{2}{1} = 2\).

Notably, the graph will never actually reach this line but will get infinitely close. This fundamentally impacts the long-term behavior of the graph and equips us with a clear end-behavior model for sketching the graph accurately.