Графік функції квадратного кореня: що таке горизонтальний перенос?

Horizontal Translations

We've learned about the base square root graph: $$y=\sqrt x$$

And we've learned about vertical translations where the graph moves along the $y$-axis.

The Horizontal Shift

The next translation is along the $x$-axis and it's the horizontal translation. Horizontal translations will be constants inside the radical. These are radicals with sums or differences. $$\boxed{y=\sqrt{x-h}}$$ Here $h$ is the horizontal shift.

• If $h$ is positive (of the form $\sqrt{x-h}$), it moves the graph to the right along the $x$-axis.
• If $h$ is negative (of the form $\sqrt{x-(-h)}$), it moves the graph to the left along the $x$-axis.

Examples$\\$ Let's look at some more specific examples.

Say we wanted to graph the equation: $$y=\sqrt{x-4}$$ We know this is a horizontal shift because the constant is inside the radical. This means we move the standard square root graph 4 spaces to the right along the $x$-axis.

We can simplify the radical by removing a common factor. $$\begin{align} y&=\sqrt{4x+20}\\ y&=\sqrt{4(x+5)}\\ y&=2\sqrt{x+5}\\ y&=2\sqrt{x-(\color{#8925AE}{-5})} \end{align}$$ The horizontal shift is $-5$.

To make this a little more complicated, check out our lesson on radical expression word problems!

Horizontal Translation of Square Root Graphs

We have already learned how to graph square roots using a $t$-table, and how to apply vertical translations to our square root graphs. Now, we are going to focus on horizontal translations.

In general terms, a square root equation will have a horizontal translation in the graph when there is an addition or subtraction inside of the radical.

$$y=\sqrt{x\pm a}$$

However, the outcome of the translation on graph doesn't exactly match what you would expect. Let's look at a graph comparing $y=\sqrt{x}$ and $y=\sqrt{x+2}$.

This is a graph of $y=\sqrt{x}$ in red and $y=\sqrt{x+2}$ in green.

The function $y=\sqrt{x+2}$ is two units to the left of the function $y=\sqrt{x}$. This may not have been what you are expecting, but the graph will move horizontally to the left when a number is added to the function under the radical. When a number is subtracted under the radical, the function will be shifted to the right. Let's look at another comparison.

The graph of $y=\sqrt{x}$ in red and $y=\sqrt{x-2}$ in green.

Here the function $y=\sqrt{x-2}$ is shifted two units to the right of the function $y=\sqrt{x}$

Horizontal transformations still apply when there are other transformations added to a square root function.

Graph of $y=2 \sqrt{x-1}$ in black, $y=-\sqrt{x-1}$ in blue, and $y=\sqrt{x-1}+2$ in red. Notice they all start at $x=1$.

Even though each of these graphs have a different vertical translation, they all start at $x=1$ and extend to the right. The vertical translator of a graph will not effect the horizontal transformations.

Sometimes the equation won't clearly show the translation and will require simplification. Let's look at a problem that has this kind of issue.

$$y=\sqrt{9x+18}$$

While this may look like a horizontal translation of $18$ units to the left, the stretch that is applied by the $9x$ changes the overall translation. This needs to be simplified before we can clearly see the translation.

$$\begin{align} y&=\sqrt{9x+18}\\ y&=\sqrt{9(x+2)}\\ y&=3\sqrt{x+2} \end{align}$$

After simplifying the equation from $y=\sqrt{9x+18}$ to $y=3\sqrt{x+2}$, it is clear that the shift is only $2$ units to the left rather than $18$.

(Video) Translations of the Square Root Graph

by mathman1024

This video by mathman1024 covers horizontal translations with square root graphs.

$\large{\text{Summary}}$

The function he graphs in this example is: $f(x)=-2\sqrt{x+7}+3$.

So the first thing we look at is the base function. In this example our base function is $y=\sqrt x$.

Next we see there's a $+7$ inside the function. This means we shift the graph to the left by 7 units.

The next transformation is the $+3$ outside the function. This tells us we need to shift the graph up by 3 units.

The $2$ being multiplied in front means we need to stretch the graph.

Finally, the negative symbol means we need to flip the graph across the $x$-axis.

The above is a graph of $y=-2\sqrt{x+7}+3$!