Графік функції кореня четвертого степеня

(Video) Identifying Graphs of Cube Root and Fourth Root Functions

by Larry Schmidt

Larry Schmidt has a video that goes through identifying fourth root functions. There are very few resources for graphing fourth root functions.

Fourth Root Graphs

In this lesson, we'll talk about fourth root graphs and their important characteristics. We'll start with the parent function, $y=\sqrt[4]{x}$.

The above is a graph of the fourth root function. The function starts at the origin, then increases positively. The function stays in the $1st$ quadrant (think: exponents!).

Some major points are: $$\begin{align} (0,0) \\ (1,1) \\ (16,2) \\ \end{align}$$

Transformations

Similar to other polynomial transformations we learned earlier, we can transform fourth root functions!

Horizontal shift: shifting the function left or right by a value, $c$. Adding $c$ shifts the function to the left, and subtracting $c$ shifts the function to the right. $$y=\sqrt[4]{x +c }$$

Vertical shift: shifting the function up or down by a value, $d$. Adding $d$ shifts the function up, and subtracting $d$ shifts the function down. $$y=\sqrt[4]{x} +d$$

Dilation: stretching or shrinking the function by a value, $a$. If $a>1$, the function gets stretch, and if $0<a<1$, the function gets compressed. $$y=a\sqrt[4]{x}$$

Reflection: reflecting the function over either the $x$-axis or the $y$-axis. Negating the entire function reflects the function over the $x$-axis, and negative just the $x$ value reflects the function over the $y$-axis. $$y=-\sqrt[4]{x} \text{ OR } y=\sqrt[4]{-x}$$

Fourth Root Graphs

The next type of radical function we are going to look at is the fourth root function!

The fourth root function is:

$$y=\sqrt[4]{x}$$

The graph looks like:

Because this function has an even exponent, all of the $x$ and $y$ values will be positive! That's why the graph is staying in the first quadrant. We have a whole lesson on domain and range, if you want to go more in depth.