Question

Постройте график функции y=x^3x+x^4 деленное на на x+x^2

Answer 1

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Answer 2

Graph of the Function

To graph the function y = (x^3x + x^4) / (x + x^2), we can start by analyzing the behavior of the function as x approaches positive and negative infinity.

As x approaches positive infinity, both the numerator and denominator of the function grow without bound. Therefore, the function approaches positive infinity as x approaches positive infinity.

As x approaches negative infinity, both the numerator and denominator of the function also grow without bound. However, the exponent of x in the numerator is odd, which means that the function will have opposite signs for large negative values of x. Therefore, the function approaches negative infinity as x approaches negative infinity.

Next, let's find the x-intercepts and y-intercepts of the function.

To find the x-intercepts, we set y = 0 and solve for x:

(x^3x + x^4) / (x + x^2) = 0

Since the numerator cannot be equal to zero (since it is a sum of two positive terms), the only way for the fraction to be equal to zero is if the denominator is equal to zero:

x + x^2 = 0

Factoring out an x, we get:

x(1 + x) = 0

This equation is satisfied when either x = 0 or x = -1. Therefore, the function has x-intercepts at x = 0 and x = -1.

To find the y-intercept, we set x = 0 and evaluate the function:

y = (0^3(0) + 0^4) / (0 + 0^2) = 0

Therefore, the function has a y-intercept at y = 0.

Now, let's plot the graph of the function.


The graph of the function y = (x^3x + x^4) / (x + x^2) has a vertical asymptote at x = 0, a vertical asymptote at x = -1, and a y-intercept at y = 0. The function approaches positive infinity as x approaches positive infinity and approaches negative infinity as x approaches negative infinity.

Please note that the graph is a rough sketch and may not be to scale.