Question

Построить график функции y=(x^2-4)/(x^2+4)

Answer 1

график функции y=(x^2-4)/(x^2+4)

Answer 2

Graph of the Function y = (x^2 - 4)/(x^2 + 4)

To graph the function y = (x^2 - 4)/(x^2 + 4), we can start by analyzing the equation and identifying any important features.

The function is a rational function, which means it is a ratio of two polynomial functions. The numerator is x^2 - 4, and the denominator is x^2 + 4.

Let's analyze the numerator and denominator separately:

Numerator: The numerator is x^2 - 4. This is a quadratic function that can be factored as (x - 2)(x + 2). The roots of the numerator are x = 2 and x = -2.

Denominator: The denominator is x^2 + 4. This is also a quadratic function, but it does not have any real roots. The minimum value of the denominator occurs when x = 0, and the value is 4.

Now, let's analyze the behavior of the function as x approaches positive infinity and negative infinity:

- As x approaches positive infinity, both the numerator and denominator grow without bound. Therefore, the function approaches 1. - As x approaches negative infinity, both the numerator and denominator grow without bound. Therefore, the function also approaches 1.

Based on this analysis, we can conclude that the function has a horizontal asymptote at y = 1.

To graph the function, we can plot a few points and connect them to get an idea of the shape of the graph. Let's choose some x-values and calculate the corresponding y-values:

- When x = -3, y = (-3^2 - 4)/(-3^2 + 4) = 5/13 ≈ 0.385 - When x = -2, y = (-2^2 - 4)/(-2^2 + 4) = 0/0 (undefined) - When x = -1, y = (-1^2 - 4)/(-1^2 + 4) = -3/3 = -1 - When x = 0, y = (0^2 - 4)/(0^2 + 4) = -4/4 = -1 - When x = 1, y = (1^2 - 4)/(1^2 + 4) = -3/5 ≈ -0.6 - When x = 2, y = (2^2 - 4)/(2^2 + 4) = 0/8 = 0 - When x = 3, y = (3^2 - 4)/(3^2 + 4) = 5/13 ≈ 0.385

Now, let's plot these points on a graph:


Based on the plotted points, we can see that the graph approaches the horizontal asymptote y = 1 as x approaches positive and negative infinity. The graph also has a vertical asymptote at x = -2 and x = 2, where the denominator becomes zero.

Please note that the graph may have additional features that are not apparent from the plotted points. To get a more accurate representation of the graph, you can use graphing software or a graphing calculator.