Question
Describe the end behavior x^4-7x^2-18=0
Answers
Answered by
GPT 3.5
The end behavior of the equation x^4 - 7x^2 - 18 = 0 describes what happens to the function as x approaches positive infinity and negative infinity.
To determine the end behavior, we first need to factor the equation:
(x^2 - 9)(x^2 + 2) = 0
From this factored form, we can see two sets of solutions:
1) x^2 - 9 = 0
(x + 3)(x - 3) = 0
x = -3, 3
2) x^2 + 2 = 0
This equation has no real solutions.
Therefore, the equation x^4 - 7x^2 - 18 = 0 has two real solutions at x = -3 and x = 3.
Now, let's examine the end behavior:
As x approaches positive infinity, the function x^4 - 7x^2 - 18 = 0 increases without bound. This means that the graph of the equation goes up infinitely in the positive y-direction.
Similarly, as x approaches negative infinity, the function x^4 - 7x^2 - 18 = 0 also increases without bound. Therefore, the graph goes up infinitely in the negative y-direction.
In summary, the end behavior of the equation x^4 - 7x^2 - 18 = 0 is an increasing function going up infinitely in both the positive and negative y-directions as x approaches positive infinity and negative infinity, respectively.
To determine the end behavior, we first need to factor the equation:
(x^2 - 9)(x^2 + 2) = 0
From this factored form, we can see two sets of solutions:
1) x^2 - 9 = 0
(x + 3)(x - 3) = 0
x = -3, 3
2) x^2 + 2 = 0
This equation has no real solutions.
Therefore, the equation x^4 - 7x^2 - 18 = 0 has two real solutions at x = -3 and x = 3.
Now, let's examine the end behavior:
As x approaches positive infinity, the function x^4 - 7x^2 - 18 = 0 increases without bound. This means that the graph of the equation goes up infinitely in the positive y-direction.
Similarly, as x approaches negative infinity, the function x^4 - 7x^2 - 18 = 0 also increases without bound. Therefore, the graph goes up infinitely in the negative y-direction.
In summary, the end behavior of the equation x^4 - 7x^2 - 18 = 0 is an increasing function going up infinitely in both the positive and negative y-directions as x approaches positive infinity and negative infinity, respectively.