Lesson 1: Graphing Square Root Functions

Property

A radical function is a function that is defined by a radical expression. To evaluate a radical function, we find the value of $f(x)$ for a given value of $x$ just as we did in our previous work with functions.

Examples

• For the function $f(x) = \sqrt{2x - 1}$, to find $f(5)$, substitute 5 for $x$: $f(5) = \sqrt{2(5) - 1} = \sqrt{9} = 3$.
• For the function $g(x) = \sqrt[3]{x - 6}$, to find $g(-2)$, substitute -2 for $x$: $g(-2) = \sqrt[3]{-2 - 6} = \sqrt[3]{-8} = -2$.
• For the function $f(x) = \sqrt[4]{5x - 4}$, evaluating $f(-12)$ gives $\sqrt[4]{-64}$, which is not a real number, so the function has no value at $x = -12$.

Explanation

A radical function is simply a function that contains a root, like a square root or cube root. To evaluate it, you just substitute the given number for the variable $x$ and then simplify the expression under the radical.

Property

The parent square root function is defined as:

Examples

Property

The graph of $f(x) = \sqrt{x} + k$ shifts the graph of $f(x) = \sqrt{x}$ vertically $k$ units.

• If $k > 0$, shift the graph vertically up $k$ units.
• If $k < 0$, shift the graph vertically down $|k|$ units.

Examples