Complete the square before restricting the graph. On a bounded interval, compare the included endpoints with the turning point if it lies in the interval. An excluded endpoint may give a limiting bound that is not attained. For y=2(x-3)2-5, 1 <= x<4, the vertex gives the minimum -5 and the included endpoint x=1 gives the maximum 3: range [-5,3]. For 4<x 6, the same rule is increasing and its range is (-3,13].

Complete the square before restricting the graph. On a bounded interval, compare the included endpoints with the turning point if it lies in the interval. An excluded endpoint may give a limiting bound that is not attained.

For \(y=2(x-3)^2-5\), \(1\le x<4\), the vertex gives the minimum \(-5\) and the included endpoint \(x=1\) gives the maximum \(3\): range \([-5,3]\). For \(4<x\le6\), the same rule is increasing and its range is \((-3,13]\).

Endpoints and turning points

Endpoints and turning points