Maths · Algebra and graphs. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
A sketch is not an accurate plot. It shows the shape of the curve and labels its important points: where it crosses each axis and where it turns.
For a quadratic y = ax2 + bx + c the curve is a parabola. It is U-shaped with a minimum point when a is positive, and upside down with a maximum point when a is negative. It has a vertical line of symmetry that passes through the turning point and lies halfway between the two roots.
For a cubic in factor form, each factor gives one crossing point on the x-axis. When the x3 term is positive the curve runs from bottom left to top right.
Signs get reversed: y = (x − 2)(x + 3) crosses the x-axis at x = 2 and x = −3, and y = (x − 3)2 + 5 turns at x = +3. Find the value of x that makes each bracket zero.
Answer: Crosses at (0, 5), (1, 0) and (5, 0); minimum point (3, −4)
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