what statement best describes the function f(x)=2x^3+2x^2-x?
Best? That is a matter of opinion. It is a polynomial, third degree.
cubic polynomial maybe?
f(x) = x (2x^2+2x-1)
To determine the best statement that describes the function f(x) = 2x^3 + 2x^2 - x, we can analyze the properties of the function.
1. Degree of the Polynomial: The highest exponent in the function is 3, which means it is a cubic function. Therefore, we can describe the statement as "f(x) is a cubic function."
2. Leading Coefficient: The leading coefficient of the function is 2. This coefficient indicates that the function opens upwards, meaning the graph has a positive slope as the x-values increase. We can describe the statement as "f(x) is a positive cubic function."
3. Zeros and Intercepts: To find the zeros or x-intercepts of the function, we set f(x) = 0 and solve for x.
2x^3 + 2x^2 - x = 0
Unfortunately, there is no simple analytical way to solve cubic equations. We can use numerical methods, such as graphing or using a calculator, to estimate the zeros of the function. Let's approximate the zeros:
Using a graphing calculator or software, we find that the approximate zeros are:
x ≈ -1.366, x ≈ -0.366, x ≈ 0.732
These are the x-values where the graph intersects the x-axis. We can describe the statement as "f(x) has three x-intercepts approximately at x ≈ -1.366, x ≈ -0.366, and x ≈ 0.732."
4. Turning Points and Extrema: To find the turning points or extrema of the function, we can take the derivative of f(x) and find where it equals zero.
f'(x) = 6x^2 + 4x - 1
Setting f'(x) = 0 and solving for x:
6x^2 + 4x - 1 = 0
Using the quadratic formula, we find that the solutions are approximately x ≈ -0.238 and x ≈ 0.722.
These are the x-values where the graph of f(x) has turning points or extrema. We can describe the statement as "f(x) has two turning points approximately at x ≈ -0.238 and x ≈ 0.722."
Combining these statements, we can describe the function f(x) = 2x^3 + 2x^2 - x as a positive cubic function with three x-intercepts approximately at x ≈ -1.366, x ≈ -0.366, and x ≈ 0.732, and two turning points approximately at x ≈ -0.238 and x ≈ 0.722.