what is the power function end behavior model of
f(x)= 3x^2-2x+1 and how do you find it?
To determine the end behavior and the power function model for the given function f(x) = 3x^2 - 2x + 1, you need to analyze the leading term of the polynomial expression.
1. Identify the leading term: In this case, the leading term is 3x^2. It is the term with the highest power of x in the polynomial.
2. Determine the degree: The degree of a polynomial is equal to the exponent of the leading term. In this case, the degree is 2 since the exponent of x is 2.
3. Analyze the sign of the leading coefficient: The leading coefficient is the coefficient of the leading term. In this case, the leading coefficient is 3.
Now, let's discuss the power function model and end behavior based on the degree and leading coefficient.
Since the degree is 2, which is an even number, the power function model can be classified into two possible forms:
1. If the leading coefficient (3) is positive, the power function model would be f(x) = ax^2, where "a" represents a positive constant. In this case, the graph of the function would open upwards (concave upward) as it approaches positive infinity and downwards (concave downward) as it approaches negative infinity.
2. If the leading coefficient (3) is negative, the power function model would be f(x) = -ax^2, where "a" represents a positive constant. In this case, the graph of the function would open downwards (concave downward) as it approaches positive infinity and upwards (concave upward) as it approaches negative infinity.
To summarize, for the given function f(x) = 3x^2 - 2x + 1, the power function end behavior model would be a quadratic function with either an upwards or downwards opening, depending on the sign of the leading coefficient.
To find the end behavior model of a power function, we look at the leading term of the function as x approaches positive and negative infinity.
In the given function f(x) = 3x^2 - 2x + 1, the leading term is 3x^2.
As x approaches positive infinity, the dominant term 3x^2 will have a positive sign. Therefore, the end behavior model can be written as f(x) ≈ 3x^2 as x → ∞.
Similarly, as x approaches negative infinity, the dominant term 3x^2 will also have a positive sign. Hence, the end behavior model can be written as f(x) ≈ 3x^2 as x → -∞.
Therefore, the power function end behavior model of f(x) = 3x^2 - 2x + 1 is f(x) ≈ 3x^2 as x → ±∞.