ACT (Enhanced)MathematicsHard
A scientist is studying the relationship between the number of hours a plant is exposed to light (h) and its growth in centimeters (g). The relationship is modeled by the function g(h) = -0.1h^2 + 2h + 5. What is the average rate of change of the plant's growth between 5 hours and 10 hours of light exposure?
- A2.0 cm/hour
- B0.5 cm/hour
- C1.0 cm/hour
- D1.5 cm/hour
Show answer & explanationAnswer & explanation
Correct answer: B. 0.5 cm/hour
The average rate of change is calculated as (g(h2) - g(h1)) / (h2 - h1). First, evaluate g(5) and g(10). Then, apply the formula.
Why the other options are wrong
- A. This might be the coefficient of the linear term, incorrectly assumed as the rate of change.
- C. This results from a calculation error in evaluating the function or the rate.
- D. This is a plausible distractor, often resulting from a slight arithmetic error.
Average Rate of Change (Quadratic)
For a quadratic function, the average rate of change over an interval [a, b] is the slope of the secant line connecting the points (a, f(a)) and (b, f(b)).
- Formula: (f(b) - f(a)) / (b - a).
- Represents the average change in the dependent variable per unit change in the independent variable.
- Different from instantaneous rate of change (derivative).
Memory trick: Y2 minus Y1, over X2 minus X1, the average change is done!