A quadratic graph is a parabola
gives a U-shape (a > 0) or ∩-shape (a < 0). It crosses the x-axis at the solutions of and the y-axis at .
Solve by reading roots from the graph
Where the curve crosses the x-axis (y = 0) are the solutions; symmetry puts the turning point exactly halfway between the two roots.
Completing the square gives the turning point
Writing shows the turning point is at — the minimum (or maximum) of the curve.
Drawn from real examiner reports.
Misreading the graph scale
Marks were lost reading values off the axes when each square was not one unit — always check the scale before reading a root or a turning point.
A recurring caution; a recent higher-tier exam asks students to "be able to read graph scales accurately".
Reading factors off a calculator
Reading factors off a calculator gave forms like (x + 2.1)(x − 1) that do not expand back to the given quadratic, and scored nothing where algebraic working was required.
Seen on a recent higher-tier exam.
No sketch for a quadratic inequality
Students who drew a sketch had a much better chance of writing the correct final inequality (single region vs two distinct regions); those who did not often gave the wrong direction.
Highlighted on recent higher-tier exams.
Forgetting to multiply −9 by the factor
When , e.g. , completing the square inside gives . You must multiply the by the : . Leaving un-multiplied is the classic error.
Turning point read as the x-value
From the minimum VALUE is (the -coordinate), reached at . Quoting as the minimum, or the turning point as instead of , both lose marks.
Plot a smooth curve and find symmetry
Join plotted points with a single smooth curve (never ruled segments) and use the line of symmetry to locate the turning point.
Solve a quadratic inequality with a sketch
Find the roots, sketch the parabola, then read the region: for an upward curve, is BETWEEN the roots and is OUTSIDE them. The sketch stops you giving the wrong region or direction.
Use the discriminant for root count
Before sketching, tells you how the curve meets the -axis: two roots, one (touches), none (curve misses the axis). Use brackets: , not .
is a parabola: a U-shape if , an ∩-shape if .
How do you read the solutions of a quadratic from its graph?
Find the coordinates of the turning point of by completing the square.
