CAIE A-Level Mathematics AS 1.1.1 Completing the Square Questions

Practise completing the square to rewrite quadratics, locate vertices, decide ranges and link a curve to a tangent horizontal line.

Exam points

rewrite quadratic expressions by completing the square
identify the vertex and range of a quadratic function
use completed-square form to solve and interpret quadratics

[Maximum number: 3]

The function f is defined by $\mathrm{f}(x)=10+6 x-x^{2}$ for $x \in \mathbb{R}$ .

By completing the square, find the range of f .

Express f(x) as: $a-(x-3)^{2}$ or $a-(3-x)^{2}$ where $a= \pm 19$ or $\pm 1$
OE

If the form $-f(x)=\left(x^{2}-6 x-10\right)$ is used the form must be returned to $f(x)=\ldots$

Completed square form must give $-x^{2}$ . Answers must come from completion of the square (not calculus or graphs).

$19-(3-x)^{2}$ or $19-(x-3)^{2}$
OE

$f(x) \leqslant 19$ or $y \leqslant 19$ with $\leqslant$ , not <
or $-\infty<\mathrm{f}(x) \leqslant 19$ or $-\infty \leqslant \mathrm{f}(x) \leqslant 19$
or $(-\infty, 19]$ or $[-\infty, 19]$

A1 FT

Using their constant following the award of M1.
SC B1 answer only or answer from a method not
involving completion of the square.

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CAIE A-Level Mathematics AS 1.1.1 Completing the Square Questions question 1