what is the power function end behavior model of
f(x)= 3x^2-2x+1 and how do you find it?
This is a quadratic function, a parabola
It has complex zeros (1/3)(1 +/- i sqrt 2) so does not cross the x axis.
What happens as x goes to +oo or - oo?
as |x| gets big, x^2 is >> x or 1 so it acts like y = 3 x^2 for large plus or minus x
So it is an upright parabola (holds water)
Let's find the vertex
3 x^2 - 2x = y-1
x^2 - 2/3 x = y/3 - 1/3
x^2 - 2/3 x + 1/9 = y/3 -2/9
(x-1/3)^2 = (1/3)(y-2/3)
vertex at (1/3, 2/3)
To determine the power function end behavior model of the given function f(x) = 3x^2 - 2x + 1, you need to consider the leading term of the polynomial, which is the term with the highest exponent. In this case, the leading term is 3x^2.
The end behavior of a power function can be determined by looking at the sign (+ or -) of the leading coefficient and the evenness or oddness of the exponent.
1. Start by determining the sign of the leading coefficient. In this case, it is positive (+3).
2. Next, consider the evenness or oddness of the exponent. Since the exponent is 2 (an even number), this means the function is an even function.
Knowing that the leading coefficient is positive and the exponent is even, we can conclude the following:
- As x approaches positive infinity (+∞), the function f(x) will also approach positive infinity (+∞).
- As x approaches negative infinity (-∞), the function f(x) will also approach positive infinity (+∞).
In summary, the power function end behavior model of f(x) = 3x^2 - 2x + 1 is that as x approaches infinity or negative infinity, the function approaches positive infinity.
To find the power function end behavior model of a given function, you need to examine the highest power term of the function.
In the given function f(x) = 3x^2 - 2x + 1, the highest power term is 3x^2.
To determine the end behavior model, follow these steps:
1. Identify the degree of the highest power term: In this case, the degree of 3x^2 is 2.
2. Determine the sign of the coefficient of the leading term: The coefficient of the leading term is 3, which is positive.
3. Determine whether the degree of the polynomial is even or odd: Since the degree is 2 (an even number), the end behavior of the function will be the same on both ends of the graph.
For an even-degree polynomial with a positive leading coefficient, the end behavior model is "As x approaches positive or negative infinity, f(x) approaches positive infinity."
Therefore, the power function end behavior model for f(x) = 3x^2 - 2x + 1 is "As x approaches positive or negative infinity, f(x) approaches positive infinity."
To find this, you need to analyze the highest power term and the sign of its coefficient to determine whether the function increases or decreases as x approaches infinity or negative infinity. In this case, the function, represented by 3x^2, increases without bound as x approaches positive or negative infinity.