Find a power function end behavior model for f(x)= (3x^3-4x^2+3x+3)/(x-3)
I don't understand what an end behavior model is
end behaviour: when x>>>>+- infinity
when x>>large, then f(x)appx=3x^2/x=3x^2
The end behavior of a function describes what happens to the function as the input (x) approaches positive infinity or negative infinity. In other words, it describes the behavior of the function towards the right or left of the graph.
To find the end behavior model for the given function f(x) = (3x^3 - 4x^2 + 3x + 3) / (x - 3), we need to simplify it and analyze the highest degree term.
First, let's perform long division to simplify the function:
3x^2 + 5x + 18
_________________________
x - 3 | 3x^3 - 4x^2 + 3x + 3
- (3x^3 - 9x^2)
_______________
5x^2 + 3x
- (5x^2 - 15x)
_______________
18x + 3
- (18x - 54)
________________
57
After performing long division, we have:
f(x) = 3x^2 + 5x + 18 + 57 / (x - 3)
As x approaches positive infinity, the highest degree term dominates the function. The highest degree term is 3x^2.
Therefore, the end behavior model for f(x) is f(x) ≈ 3x^2 as x approaches positive infinity.
Similarly, as x approaches negative infinity, the highest degree term 3x^2 still dominates the function.
Therefore, the end behavior model for f(x) is f(x) ≈ 3x^2 as x approaches negative infinity.
An end behavior model refers to how a function behaves as the input values approach positive or negative infinity. In other words, it describes the trend of the function's values at the far ends of the graph.
To find the end behavior model for a function, we need to analyze the highest degree term in the numerator and the highest degree term in the denominator. In this case, the highest degree term in the numerator is 3x^3, and the highest degree term in the denominator is x.
To determine the end behavior of the function, we compare the degrees of the numerator and denominator terms:
- If the degree of the numerator is greater than the degree of the denominator, the end behavior will be represented by a power function.
- If the degree of the numerator is equal to the degree of the denominator, the end behavior will be represented by a horizontal asymptote.
- If the degree of the numerator is less than the degree of the denominator, the end behavior will be represented by a horizontal line.
In this case, the degree of the numerator (3) is higher than the degree of the denominator (1), so the end behavior will be represented by a power function.
Now, let's determine the end behavior model for the given function, f(x) = (3x^3 - 4x^2 + 3x + 3)/(x - 3):
1. Divide each term in the numerator by the term in the denominator to simplify the function:
f(x) = (3x^3 - 4x^2 + 3x + 3)/(x - 3)
= (3x^2 + 5x + 18) + 57/(x - 3)
2. The simplified equation is f(x) = 3x^2 + 5x + 18 + 57/(x - 3).
The end behavior model for this function is a power function represented by the term with the highest degree, which is 3x^2. Therefore, the end behavior model for f(x) is:
f(x) ≈ 3x^2 as x approaches infinity or negative infinity.