Concavity Model Justifications AP Precalculus

Concave up — on interval

Since the graph of f(x) is concave up, the secant line on the interval xA <= x <= xB lies above the curve on that interval, so using it to approximate a value within that interval will give a value greater than the value found from the model.

Concave up —outside the interval

Since the graph of f(x) is concave up, the secant line on the interval xA <= x <= xB lies below the curve for x < xA and x > xB, and thus gives values less than the value of the model on those intervals.

Concave down —on interval

Since the graph of f(x) is concave down, the secant line on the interval xA <= x <= xB lies below the curve on that interval, so using it to approximate a value within that interval will give a value less than the value given by the model.

Concave down — outside interval

Since the graph of f(x) is concave down, the secant line on the interval xA <= x <= xB lies above the curve for x < xA and x > xB, and thus yields values greater than the value of the model on those intervals.