Vertical Translation of Square Root Graphs - Definition

Vertical Translations

We've already learned about the base square root graph. $$y=\sqrt{x}$$

Now, we're going to look at vertical translations. In other words, shifting the graph up and down. The generic formula is: $$\boxed{y=\sqrt{x}+c}$$ Here, $c$ is the vertical shift.

• If $c$ is positive the graph moves upward along the $y$-axis.
• If $c$ is negative the graph moves downward along the $y$-axis.

Let's look at some specific examples.

Say we wanted to graph the equation: $$y=\sqrt x+3$$ We know this is a vertical shift because the constant is outside of the square root. The $+3$ tells us we shift the standard radical graph up $3$ spaces along the $y$-axis.

First, we want to solve for $y$. To do this we subtract $2$ from both sides and then multiply both sides by $2$. $$\begin{align} \frac y2+2&=\sqrt x\\ \frac y2&=\sqrt x -2\\ y&=2(\sqrt x-2)\\ y&=2\sqrt x\, \color{#8925AE}{-\,4} \end{align}$$ The vertical shift is $4$ units downward. Therefore, $-4$ is our answer.

To do more practice, but with word problems, check out our lesson on word problems with square roots!

Applying a vertical translation to a graph drags it up or down along the $y$-axis. The amount of this translation is a constant, $c$, that we get to pick, and it can be positive or negative. We add or subtract a constant outside of a function to get another function that takes different values from the original one.

This notation works for any function, but we want to focus on translating square root functions. Let's begin with a graph of the simplest of all square root functions, $f(x) = \sqrt{x}.$

Let's see an example of a vertical translation.

What we have here is a graph of $f(x) = \sqrt{x} + 4.$ The $+4$ means that we are shifting the graph upward by $4$ units. A downward shift will involve subtracting a term.

This is a graph of $f(x) = \sqrt{x} - \frac{3}{2}.$ We can replace the $-\frac{3}{2}$ with any real number we want and still have a vertical translation of $\sqrt{x}.$ Let's see an example of a graph that is NOT a vertical translation of the square root function.

This is actually a horizontal translation of the graph of $\sqrt{x}.$ Notice we are dragging $\sqrt{x}$ to the left rather than up or down.

As a final comment, no matter what vertical translation of $\sqrt{x}$ we use, the new function will still be undefined for negative inputs.

We can translate other functions besides square root functions, but those get more complicated!

(Video) Vertical Translation of Square Root Graphs

by Eric Stanwood

This video by Eric Stanwood demonstrates how to create a vertical translation of a square root function.

Summary

Here is the graph of a standard parent square root function, $y=\sqrt{x}$:

If we want to shift the function up by one unit, we add $1$ outside of the radical sign.

This is a graph of the function $y=\sqrt{x}+1$:

Now, let's look at the parent function again, $y=\sqrt{x}$:

Let's shift this parent function down by one unit. We are subtracting one from outside of the radical sign.

This is a graph of that function, $y=\sqrt{x}-1$: