A radical function is a function that contains a radical expression with a variable in the radicand.
The simplest radical function is the square root function:
\$f(x) = \sqrt{x}\$.
Note
The domainof the square root functionis $[0, \infty)$, since the square root of a negative number is not defined in the real numbers.
Placeholder(graph): Graph of the square root function f(x) = sqrt(x)
Note
The transformationsof radical functionsare similarto those of otherfunctions.
Horizontal Shifts
The function \$f(x) = \sqrt{x - h}\$ is the square root function \$f(x) = \sqrt{x}\$ shifted horizontally by \$h\$ units:
- If \$h > 0\$, the graph is shifted to the right.
- If \$h < 0\$, the graph is shifted to the left.
Placeholder(graph): Graph of the function f(x) = sqrt(x - 2), showing a horizontal shift of 2 units to the right
Placeholder(graph): Graph of the function f(x) = sqrt(x) + 3, showing a vertical shift of 3 units up
Placeholder(graph): Graph of the function f(x) = -sqrt(x), showing a reflection across the x-axis
Placeholder(graph): Graph of the function f(x) = 2sqrt(x), showing a vertical stretch by a factor of 2
Placeholder(graph): Graph of the function f(x) = sqrt(2x), showing a horizontal compression by a factor of 2
Placeholder(graph): Graph of the function f(x) = 2sqrt(x - 3) + 1, showing the transformations applied to the parent function f(x) = sqrt(x)
Self review
1. Graphthe function \$f(x) = -\sqrt{x + 2} - 3\$. 2. Graphthe function \$f(x) = \frac{1}{2}\sqrt{x - 1} + 4\$.
