# Graphs of Square Root Functions

Graphing Square Root Functions
11:07 — by Marty Brandl

Tip: This isn't the place to ask a question because the teacher can't reply.

## Key Questions

• graph{sqrt(x) [-10, 10, -5.21, 5.21]}
The parent graph of the square root function looks like this.
To find the points on the graph, plug in the x value into the square root to find the y-value.
For example at x-value 1, the y-value is also one.
At x-value 4, the y-value is 2.
This is the parent graph of $y = \sqrt{x}$

When there is other numbers in the equation, the numbers inside of the square root move the graph right and left. For example $y = \sqrt{x - 2}$ would look like graph{sqrt(x-2) [-10, 10, -5.21, 5.21]}

If the number is outside of the square root, it moves it up and down.
For exampel $y = \sqrt{x} + 2$ would look like graph{sqrt(x)+2 [-10, 10, -5.21, 5.21]}

This is the basics on how to draw square root graphs.

• In order to translate any function to the right or left, place an addition or subtraction "inside" of the Parent function. In the case of the square root function, it would look like y = $\sqrt{x - 2}$ or y = $\sqrt{x + 5}$.

Let's look at the effect of the addition or subtraction. First, the domain will be altered. You must set x - 2 $\ge$ 0 , or say that you understand that the square root function has a domain of $x \ge 2$.
This implies a horizontal shift/translation of 2 units to the right. (see graph)

Now repeat for x + 5 $\ge$ 0, or $x \ge - 5$. This graph will be translated 5 units to the left. (see graph)

Now, let's explore how to translate a square root function vertically. y = $\sqrt{x} + 3$ or y = $\sqrt{x} - 4$. The addition or subtraction on the OUTSIDE of the square root function will cause the graph to translate up or down. Adding 3 will raise the graph up, and subtracting 4 will lower the graph by 4 units. (see graph)

If you are ready for a challenge, we can try to translate in more than one direction at a time!

y = $\sqrt{x + 2} - 7$ First, make a prediction...will the graph be translated right 2 or left 2, up 7 or down 7?

Find the domain by setting x + 2 $\ge$ 0 for starters.
$x \ge - 2$ (that means left 2, of course!)
And, subtraction of 7, must mean down 7. (see graph)

• Graph them like quadratic functions.
However, since square root cannot be performed for negative numbers, you need only graph the "positive" part of the quadratic.

• you have half of a parabola.

Consider $y = \sqrt{x}$

$x = 0 \implies y = 0$
$x = 1 \implies y = 1$
$x = 4 \implies y = 2$
$x = 9 \implies y = 3$
$x = - 1 \implies$ Undefined in $\mathbb{R}$

You have upper part of a parabola that opens to the right

If you consider $y = - \sqrt{x}$

You have the lower part of a parabola that opens to the right.

$\sqrt{y} = x$ and $- \sqrt{y} = x$ behaves similarly

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