Think of steeper hills and faster downhill rides! The steeper parabola (with the bigger negative coefficient) changes more rapidly (has a bigger average rate of change) than the shallower one, even though they both drop downwards. So, g(x) takes a steeper plunge than f(x) within the given interval!
To compare the average rates of change for the functions f(x) = -0.5x^2 and g(x) = -1.5x^2 over the interval -5 ≤ x ≤ -2:
1. Absolute Value of Coefficients:
Notice that the coefficients in front of the squared terms differ: -0.5 for f(x) and -1.5 for g(x). Since both coefficients are negative, they represent downward-facing parabolas.
The absolute value of the coefficient determines the steepness of the parabola. In this case, |-1.5| > |-0.5|, meaning g(x) has a steeper slope than f(x).
2. Average Rate of Change:
The average rate of change over an interval measures the overall change in the function's output relative to the change in the input. For parabolas, the average rate of change is calculated as the difference in y-values at the interval's endpoints divided by the difference in x-values.
Since both parabolas open downwards, their average rates of change will be negative.
3. Comparison:
Given the steeper slope of g(x), its absolute value of the average rate of change will be greater than the absolute value of the average rate of change for f(x). In other words, g(x) will change more rapidly (in absolute terms) than f(x) over the given interval.
Therefore, g(x) has a steeper average rate of change compared to f(x) over the interval -5 ≤ x ≤ -2. This means g(x)'s output value changes more rapidly (gets further from the origin) as the input value increases from -5 to -2.
Visualization:
Imagine two parabolas, one shallower (f(x)) and one steeper (g(x)), both opening downwards within the specified interval. The steeper parabola (g(x)) will trace a path further away from the origin compared to the shallower one (f(x)) within that interval, indicating a larger average change in its output.