Recognise linear, quadratic and cubic shapes

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Recognise linear, quadratic and cubic shapes

A rule determines the shape and important features of its graph. A nonvertical straight line has constant gradient. The graph y = x² is a U-shaped parabola through the origin, with nonnegative outputs. The graph y = x³ passes through the origin and takes negative outputs for negative inputs.

Tables help locate points, but a sketch also needs the behaviour between them. For y = x³, inputs −2, −1, 0, 1, 2 give outputs −8, −1, 0, 1, 8. The curve increases throughout; its flattening near the origin is not a maximum or minimum.

Symmetry helps check a drawing. The graph of x² is reflected onto itself in the y-axis, whereas x³ has rotational symmetry about the origin. A graph-family sketch communicates these features without pretending to provide exact readings everywhere.

Worked example: Calculate points for y = x³ at x = −2, 0, 2.

Cube each signed input.

An odd number of negative factors gives a negative result.

Answer: (−2, −8), (0, 0), (2, 8).

Check your understanding: Can y = x² have output −4 for a real input?

Answer: No.

Why: The square of a real number is nonnegative.