Final answer:
The turning points of the given function are (-4, -48) and (-2, -20). The function is increasing on the intervals (-∞, -4) and (-2, ∞), and decreasing on the interval (-4, -2). The parent function is f(x) = x^3, and the range of the function is all real numbers. The zeros of the function are x = 0, x = -2, and x = -4.
Step-by-step explanation:
Turning Points:
To find the turning points, we need to find the derivative of the function and set it equal to zero. Differentiating f(x) = x^3 + 6x^2 + 8x, we get f'(x) = 3x^2 + 12x + 8. Setting f'(x) = 0 gives us x = -4 and x = -2. So, the turning points are (-4, -48) and (-2, -20).
Increasing and Decreasing Intervals:
To determine the increasing and decreasing intervals, we need to analyze the sign of the derivative. We can see that the derivative is positive for x < -4, negative between -4 and -2, and positive for x > -2. Therefore, the function is increasing on the intervals (-∞, -4) and (-2, ∞), and decreasing on the interval (-4, -2).
Parent Function:
The parent function is the function without any transformations or changes. In this case, the parent function would be f(x) = x^3, since all the additional terms and coefficients are considered transformations.
Range:
The range of the function f(x) = x^3 + 6x^2 + 8x is all real numbers, since the cubic function can take any y-value.
Zeros:
To find the zeros or x-intercepts of the function, we set f(x) = 0 and solve for x. In this case, we have x^3 + 6x^2 + 8x = 0. Factoring out an x, we get x(x^2 + 6x + 8) = 0. Setting each factor equal to zero, we find x = 0, x = -2, and x = -4 as the zeros of the function.