Transformations of Radical Functions

A radical function like y = √x can be shifted, stretched, or reflected the same way as any function, by changing what happens inside and outside the radical. Once you know the transformation rules, you can sketch a new radical curve without plotting individual points.

For y = a√(b(x − h)) + k, applied to the base square root function:

• h shifts the graph horizontally — right if h is positive, left if h is negative.
• k shifts the graph vertically — up if k is positive, down if k is negative.
• a stretches (|a| > 1) or compresses (|a| < 1) vertically; a negative a reflects across the x-axis.
• b stretches or compresses horizontally; a negative b reflects across the y-axis.

For y = √(x − 2) + 1, compare to y = √x: h = 2 shifts the graph right 2 units, and k = 1 shifts it up 1 unit. The starting point of the curve moves from (0, 0) to (2, 1), and the domain shifts along with it, from x ≥ 0 to x ≥ 2.

These same rules apply to any basic radical function, and once you can transform one, you can solve radical equations that involve a transformed version.

Because the domain of the base function starts where the expression under the radical is non-negative, any horizontal shift moves that starting point too. Always solve the inequality inside the radical ≥ 0 for the transformed function rather than assuming the domain is unchanged.

Related lessons

• Algebra