Question

over what interval is the function increasing, and over what interval is the function decreasing?
the function f(x) is increasing over the interval x > 0
(simplify your answer. type an inequality.)
the function f(x) is decreasing over the interval
(simplify your answer. type an inequality.)

Explanation:

Step1: Analyze the function's behavior

The function given is \( f(x) = 0.8x^2 \), which is a quadratic function with a positive leading coefficient (0.8 > 0), so its graph is a parabola opening upwards. The vertex of the parabola \( y = ax^2 + bx + c \) (here \( b = 0, c = 0 \)) is at \( x = -\frac{b}{2a}=0 \). For a parabola opening upwards, the function decreases to the left of the vertex (x < 0) and increases to the right of the vertex (x > 0). We can also verify from the table: when x increases from -2 to 0 (x values: -2, -1, 0), the f(x) values are 3.2, 0.8, 0, which are decreasing. When x increases from 0 to 2 (x values: 0, 1, 2), the f(x) values are 0, 0.8, 3.2, which are increasing.

Step2: Determine the decreasing interval

From the analysis (both the shape of the quadratic function and the table values), the function is decreasing when x is less than 0. So the interval where the function is decreasing is \( x < 0 \).

Answer:

The function \( f(x) \) is decreasing over the interval \( \boldsymbol{x < 0} \)