Graphing Radical Functions

The graph of y = √x starts at the origin and rises to the right, forever slower. It is the top half of a sideways parabola.

Domain

A square root refuses negative inputs, so the domain of y = √x is x ≥ 0 and the range is y ≥ 0.

Shifted domains

For y = √(x − 3) the inside must be at least zero, so the domain is x ≥ 3 and the graph starts there.

Same rules as everything else

The transformations are the ones from Chapter 1. Outside is vertical, inside is horizontal and backwards.

Cube roots differ

A cube root accepts negatives, since (−2)³ = −8. Its domain and range are all real numbers, and its graph passes through the origin in an S shape.

Why the difference

Squaring destroys the sign, so a square root cannot recover one. Cubing keeps it, so a cube root can.

Half a parabola on its side

y = √x is the upper half of the sideways parabola x = y². The square root symbol means the principal root, so only the non-negative half appears — otherwise it would fail the vertical line test.

The domain is restricted

y = √(x − 3) requires x ≥ 3, since the inside cannot be negative. Solving the inside-is-non-negative inequality gives the domain, and it determines where the graph starts.

Transformations work as usual

y = −2√(x − 3) + 1 shifts right 3, stretches by 2, reflects and raises 1. The general transformation rules apply to radical graphs exactly as to any other, which is the point of learning them generally.

Cube roots behave differently

y = ∛x accepts negative inputs, so its domain is all reals and its graph runs through all four relevant quadrants. Odd roots have no domain restriction, which is the same asymmetry seen with cube roots in Grade 8.