Writing a quadratic as a(x + h)² + k, finding its turning point, range and axis of symmetry, and sketching with intercepts.
4 narrated scenes. Press play, listen and read along — the points appear as they are explained. Use ‹ Prev / Next › or the scene bar to jump to any scene.
A quadratic hides its maximum or minimum in plain sight. Completing the square pulls it out — no calculus needed.
Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.
Change a, b and c and watch the turning point and the roots.
Copy these into your notebook in your own words — that is how they stick.
ax² + bx + c = a[(x + b/2a)² − b²/4a²] + c = a(x + h)² + k with h = b/2a, k = c − b²/4a.
(−h, k); axis of symmetry x = −h.
a > 0: minimum value k; a < 0: maximum value k.
a > 0: f(x) ≥ k; a < 0: f(x) ≤ k. Check end points for a restricted domain.
y-intercept c; x-intercepts from f(x) = 0; turning point; axis of symmetry; shape ∪ or ∩.
If the minimum k > 0 (a > 0) or the maximum k < 0 (a < 0).
Expand your a(x + h)² + k to recover ax² + bx + c.
Profit, area and height problems: complete the square to find the optimum.
3 quick games on this lesson. Earn XP and badges as you play — your best score for each game is saved. Games are for practice and do not change your exam readiness. More games in the Additional Mathematics Arcade.
8 items modelled on real CSEC Paper 1 questions. Choose one answer for each, then check. Aim for 80% or better. Work each item on paper first.
Write full answers, then reveal the mark scheme and tick the points you earned (24 marks in total). Show all your working — method marks count. Your answers are saved on this device.
Question 1 [13 marks]
f(x) = 3x² − 9x + 1.
(a) Express f(x) in the form a(x + h)² + k, where a, h and k are real numbers. [3]
(b) State the coordinates of the minimum point of f(x) and the range of f(x). [3]
(c) The equation 3x² − 9x + 1 = 0 has roots α and β. Find the value of α + β and αβ, and hence α² + β². [4]
(d) Sketch the graph of y = f(x), showing the y-intercept and the turning point. [3]
Question 2 [11 marks]
A farmer has 60 m of fencing to enclose a rectangular pen against a straight wall. The side parallel to the wall needs no fencing. Let the width of the pen be x m.
(a) Show that the area is A = x(60 − 2x) = 60x − 2x². [2]
(b) Express A in the form a(x − h)² + k. [3]
(c) Hence state the maximum area and the dimensions of the pen that give it. [3]
(d) State the range of values of x for which the pen has an area of at least 400 m². [3]
Find these in your own past-paper booklets for extra timed practice. Items above are written in the style of these papers; any adapted from a specific question say so.