A restricted quadratic model
A class uses the function f(x) = x² - 6x + 13 to represent a fictional index. The input is restricted to 0 ≤ x ≤ 5; the index has no claimed physical unit or real-world measurement. One student expands and factors different expressions, while another rewrites the function as (x - 3)² + 4. They must determine the smallest and largest allowed values, then explain why the same procedure would need an additional check if the interval changed. The task is not to read values from an omitted graph: every algebraic fact needed is given here. A calculator is unnecessary, and no assumption about official test permissions is implied. Record both the input producing each extreme and the corresponding output.
1. Find the minimum and maximum on the stated interval.
Read the worked answer
The vertex x=3 lies in the interval and gives the minimum4. The endpoints give f(0)=13 and f(5)=8, so the maximum is13 at x=0.
2. What changes if only 4 ≤ x ≤ 5 is allowed?
Read the worked answer
The vertex is excluded. On this interval the function increases, so checking the endpoints gives minimum5 at x=4 and maximum8 at x=5.