Графік функції квадратного кореня

Square Root Graphs

We're going to dive into the details of the square root graph. Let's start by looking at the most basic case, $f(x)=\sqrt{x}$.

The graph starts at the point $(0,0)$ and extends upwards and to the right. Written in interval notation: $$\begin{align} \text{Domain}&: [0,\infty)\\ \text{Range}&: [0,\infty)\end{align}$$

This is because of the rules for both square roots and functions. We know that we cannot have a negative underneath the square root (such as $\sqrt{-3}$). This means that we will only have positive $x$-values in an unaltered square root function. We also know that a function needs to pass the vertical line test, or in other words, there can only be $1$ $y$-value for every $x$-value. This means that we will only use the positive answers for a square root. For example, we will only use $\sqrt{9}=3$ when it comes to graphing the unaltered square root function.

Transformations

Square root functions can be transformed the same way any other function can. These can be categorized into shifts, stretches/compressions, and reflections.

Shifts

Shifts can be shown by adding or subtracting. Let's take a look at the graph of the following functions (with matching colors): $$\begin{alignat*}{2} f(x)&=\sqrt{x} &&\text{ Original}\\ f(x)&=\color{blue}{\sqrt{x}+1} &&\text{ Shift up one unit}\\ f(x)&=\color{green}{\sqrt{x}-1} &&\text{ Shift down one unit}\\ f(x)&=\color{purple}{\sqrt{x-1} }&&\text{ Shift right one unit}\\ f(x)&=\color{red}{\sqrt{x+1}} &&\text{ Shift left one unit}\end{alignat*}$$

The above graph shows all of the possible directions of shifts from the parent function $f(x)=\sqrt{x}$.

Stretch/Compress

Stretches/Compressions are done by multiplying. The following functions show a stretch or compression: $$\begin{alignat}{2}f(x)&=a\sqrt{x} &&\text{ Vertical stretch/compression}\\ f(x)&=\sqrt{ax} &&\text{ Horizontal stretch/compression}\end{alignat}$$

Let's look at the following examples on a graph (with matching colors): $$\begin{alignat}{2} f(x)&=\sqrt{x} &&\text{ Original}\\ f(x)&=\color{blue}{2\sqrt{x}} &&\text{ Vertical stretch by }2\\ f(x)&=\color{green}{\frac12 \sqrt{x}} &&\text{ Vertical compression by }2\\ f(x)&=\color{purple}{\sqrt{\frac12 x}} &&\text{ Horizontal stretch by } 2\\ f(x)&=\color{red}{\sqrt{2x}}&&\text{ Horizontal compression by }2\end{alignat}$$

Notice that the vertical stretch and compression are the extremes. Horizontal stretches and compressions can be a little bit hard to visualize, but they also have a small vertical component when looking at the graph.

Reflction

Reflections are the most clear on the graph but they can cause some confusion. Let's look at the following functions on the graph: $$\begin{alignat}{2}f(x)&=\sqrt{x} &&\text{ Original}\\ f(x)&=\color{blue}{-\sqrt{x}} &&\text{ Vertical reflection}\\ f(x)&=\color{green}{\sqrt{-x}} &&\text{ Horizontal reflection}\\ f(x)&=\color{purple}{-\sqrt{-x}} &&\text{ Vertical and Horizontal reflection} \end{alignat}$$

The reflections allow the square root graph to go in any direction!

Combinations of Transformations

A quick note about horizontal reflections: reflecting over the $y$-axis makes the horizontal shifts behave the opposite of what you've seen before. Based on the given information it would be easy to think that $y=\sqrt{-x}-1$ has a graph that is reflected over the $y$-axis and shifted one unit to the right, but it is actually shifted one unit to the left.

We can make some sense of this just by rearranging the function. If we pull out a $-1$ under the radical, we see something that lines up a little better with the graph: $$\begin{align}y&=\sqrt{-x-1}\\ y&=\sqrt{-1(x+1)}\end{align}$$ Now we can see the reflection and the shit as two separate parts! Always be sure to watch out for this when dealing with horizontal reflections.

We use this same method when thinking about transformations that include horizontal stretches/compressions. Try out this problem:

This function shows a vertical reflection, a horizontal stretch by a factor of $2$, a horizontal shift $\frac32$ units to the right, and a vertical shift $4$ units upwards. Let's rewrite the function to see the horizontal transformations a little more clearly, then graph it. $$\begin{align}y&=-\sqrt{2x-3}+4\\ y&=-\sqrt{2\left(x-\frac32\right)}+4\end{align}$$

Now we have the knowledge to make any square root graph! Having a good grasp on the various transformations will help with visualizing and graphing square root functions. You may get stuck or confused at some point, but you can always plug in $x$-values to find the corresponding $y$-values and graph with points!

(Video) Graphing Square Root Functions

by Marty Brandl

Marty Brandl has a very good video on how to Graph a square root function.

Square Root Graphs

We've seen function transformations before. We can do the same vertical shifts, horizontal shifts, vertical stretches and compressions, and horizontal stretches and compressions.

We've previously learned that the parent function of the square root graph is:

$$y = \sqrt{x}$$

Some characteristics of the square root parent function:

• Starts at the origin, $(0,0)$
• Curves slightly upward and to the right
• Gets less steep as $x$ increases

Some important points to remember:

• $\sqrt{0} = 0$
• $\sqrt{1} = 1$
• $\sqrt{4} = 2$
• $\sqrt{9} = 3$

So, the graph passes through: $(0,0), (1,1), (4,2), (9,3)$

These rules work for square roots of a positive integer, not necessarily the square root of negative values. That is where it gets complex.

Domain and Range

Square roots have a key restriction. You cannot take the square root of a negative number (in basic algebra).

So when we think of the domain and range:

• Domain: $x \geq 0$
• Range: $y \geq 0$

That is why the graph only appears on the right side of the coordinate plane.

Transformations

You can change the shape and position of the graph using this form:

$$y = a \sqrt{x-h} + k$$

The variables mean:

• $h \rightarrow$ shifts the graph left or right
• $k \rightarrow$ shifts the graph up or down
• $a \rightarrow$ stretches, shrinks, or reflects the graph

The starting point of a transformed graph is $(h,k)$.