Range of a Quadratic on a Restricted Domain

Range questions look simple until the domain is restricted. On the whole real line an upward parabola just needs its vertex. On a closed interval you must weigh the vertex against both endpoints, because the highest value now lives at an end.

A function is defined by $f(x) = x^2 - 4x + 7$. (i) Express $f(x)$ in completed-square form and hence state the range of $f$ for $x \in \mathbb{R}$. [3] (ii) State the range of $f$ for the restricted domain $0 \le x \le 3$. [3]

The working

(i) Complete the square (half of $-4$ is $-2$): $f(x) = x^2 - 4x + 7 = (x - 2)^2 - 4 + 7 = (x - 2)^2 + 3 \quad \text{(M1, A1)}$

Since $(x - 2)^2 \ge 0$, the smallest value is $3$ (at $x = 2$), with no upper bound over all reals: $\text{range: } f(x) \ge 3 \quad \text{(A1)}$

(ii) The vertex sits at $x = 2$, which is inside $[0, 3]$, so the minimum still occurs there: $f(2) = 3 \quad (\text{minimum on the interval})$

For the maximum, test both endpoints (M1 for evaluating endpoints): $f(0) = 0 - 0 + 7 = 7, \qquad f(3) = 9 - 12 + 7 = 4$

The larger endpoint value is $f(0) = 7$, so on this interval: $3 \le f(x) \le 7 \quad \text{(A1, A1)}$

Where the marks are won and lost

• Deciding whether the vertex lies inside the interval is the key judgement. Here $x = 2 \in [0, 3]$, so it supplies the minimum. If the vertex were outside, the minimum would move to the nearer endpoint.
• The maximum needs both endpoints checked. Testing only $x = 3$ gives $4$ and misses the true maximum of $7$ at $x = 0$.
• The full-domain range is one-sided ($f(x) \ge 3$); the restricted-domain range is a closed interval $3 \le f(x) \le 7$. Do not carry the “no upper bound” answer into part (ii).

Common mistakes

• Quoting $f(x) \ge 3$ for the restricted domain too (forgetting the interval caps it above).
• Checking only one endpoint, or neither.
• Assuming the maximum is at the right-hand endpoint $x = 3$; here it is at the left, $x = 0$.

Full method: Range of a Quadratic notes. Topic home: Quadratic Functions pillar.