Lesson 3: Graphing Radical 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

Examples

Property

The cube root parent function is $f(x) = \sqrt[3]{x}$ with domain: all real numbers and range: all real numbers.

Examples

Property

For radical functions in the form $f(x) = a\sqrt{x - h} + k$:

• Parameter $a$ creates vertical stretch (if $|a| > 1$) or compression (if $0 < |a| < 1$), and reflection over x-axis (if $a < 0$)
• Parameter $h$ creates horizontal translation: left $h$ units (if $h < 0$) or right $h$ units (if $h > 0$)
• Parameter $k$ creates vertical translation: down $k$ units (if $k < 0$) or up $k$ units (if $k > 0$)

Examples