Quadratic and other graphs

Maths · Algebra. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

The graph of a quadratic, y = ax² + bx + c, is a symmetrical curve called a parabola. It is U-shaped when a is positive and an upside-down U when a is negative. Its roots are the x-values where it crosses the x-axis (where y = 0). Its turning point is the lowest or highest point, and it lies on the line of symmetry, halfway between the roots.

Completing the square gives the turning point without drawing the graph.

You should also know other shapes by sight: cubic, reciprocal, exponential, and the graphs of sin x, cos x and tan x. A graph can be moved or reflected by changing its equation in a simple way.

Rules to remember

Roots: solve y = 0. y-intercept: put x = 0.
Completing the square: x² + bx + c = (x + b/2)² − (b/2)² + c.
y = (x + p)² + q has its turning point at (−p, q). Its x-coordinate is also halfway between the two roots.
Reciprocal y = k/x: two separate branches that never touch the axes. Exponential y = a^x: passes through (0, 1) and never touches the x-axis.
sin x and cos x stay between −1 and 1 and repeat every 360°. tan x repeats every 180°. sin x = sin (180° − x) and cos x = cos (360° − x).
Translations: y = f(x) + a moves the graph up a. y = f(x + a) moves it left a (y = f(x − a) moves it right a).
Reflections: y = −f(x) reflects in the x-axis (y-values change sign). y = f(−x) reflects in the y-axis (x-values change sign).

Common mistake

Getting the sign of the x-coordinate wrong. y = (x − 3)² + 2 has its turning point at (3, 2), not (−3, 2). In the same way, y = f(x + 3) moves the graph 3 to the left, not to the right.

Worked example

Find the turning point and the exact roots of y = x² + 8x + 10.

Complete the square: x² + 8x + 10 = (x + 4)² − 16 + 10 = (x + 4)² − 6.
Turning point: (−4, −6).
Roots: (x + 4)² − 6 = 0, so (x + 4)² = 6 and x + 4 = ±√6.
x = −4 + √6 or x = −4 − √6.

Answer: Turning point (−4, −6); roots x = −4 ± √6

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