A radical function like the square root function transforms with the same rules as any function: shifts, stretches, and reflections. Learn the transformation rules for y=a times the square root of b(x-h) plus k, with a worked example shifting the base function right 2, up 1.
Transforming a radical function
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.
y = √(x−2) + 1 is y = √x shifted 2 units right and 1 unit up.
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.
Worked example
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.
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.