What if you had a square root function like \begin{align*}y = \sqrt{2x} + 3\end{align*}

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CK-12 Foundation: Graphs of Square Root Functions

### Guidance

In this chapter you’ll learn about a different kind of function called the square root function. You’ve seen that taking the square root is very useful in solving quadratic equations. For example, to solve the equation \begin{align*}x^2 = 25\end{align*}

A square root function is any function with the form: \begin{align*}y = a \sqrt{f(x)} + c\end{align*}

**Graph and Compare Square Root Functions**

When working with square root functions, you’ll have to consider the domain of the function before graphing. The domain is very important because the function is undefined when the expression inside the square root sign is negative, and as a result there will be no graph in whatever region of \begin{align*}x-\end{align*}

To discover how the graphs of square root functions behave, let’s make a table of values and plot the points.

#### Example A

*Graph the function* \begin{align*}y = \sqrt{x}\end{align*}

**Solution**

Before we make a table of values, we need to find the domain of this square root function. The domain is found by realizing that the function is only defined when the expression inside the square root is greater than or equal to zero. Since the expression inside the square root is just \begin{align*}x\end{align*}

This means that when we make our table of values, we should pick values of \begin{align*}x\end{align*}

\begin{align*}x\end{align*} |
\begin{align*}y=\sqrt{x}\end{align*} |
---|---|

0 | \begin{align*}y=\sqrt{0} = 0\end{align*} |

1 | \begin{align*}y=\sqrt{1} = 1\end{align*} |

2 | \begin{align*}y=\sqrt{2} = 1.4\end{align*} |

3 | \begin{align*}y=\sqrt{3} = 1.7\end{align*} |

4 | \begin{align*}y=\sqrt{4} = 2\end{align*} |

5 | \begin{align*}y=\sqrt{5} = 2.2\end{align*} |

6 | \begin{align*}y=\sqrt{6} = 2.4\end{align*} |

7 | \begin{align*}y=\sqrt{7} = 2.6\end{align*} |

8 | \begin{align*}y=\sqrt{8} = 2.8\end{align*} |

9 | \begin{align*}y=\sqrt{9} = 3\end{align*} |

Here is what the graph of this table looks like:

The graphs of square root functions are always curved. The curve above looks like half of a parabola lying on its side, and in fact it is. It’s half of the parabola that you would get if you graphed the expression \begin{align*}y^2 = x\end{align*}

Notice that if we graph the two separate functions on the same coordinate axes, the combined graph is a parabola lying on its side.

Now let's compare square root functions that are multiples of each other.

#### Example B

*Graph the functions* \begin{align*}y = \sqrt{x}, y=2\sqrt{x}, y=3\sqrt{x},\end{align*}*and* \begin{align*}y = 4\sqrt{x}\end{align*}*on the same graph.*

**Solution**

Here is just the graph without the table of values:

If we multiply the function by a constant bigger than one, the function increases faster the greater the constant is.

#### Example C

*Graph the functions* \begin{align*}y=\sqrt{x}, y= \sqrt{2x}, y=\sqrt{3x},\end{align*}*and* \begin{align*}y=\sqrt{4x}\end{align*} *on the same graph.*

**Solution**

Notice that multiplying the expression *inside* the square root by a constant has the same effect as multiplying by a constant *outside* the square root; the function just increases at a slower rate because the entire function is effectively multiplied by the square root of the constant.

Also note that the graph of \begin{align*}\sqrt{4x}\end{align*} is the same as the graph of \begin{align*}2\sqrt{x}\end{align*}. This makes sense algebraically since \begin{align*}\sqrt{4} = 2\end{align*}.

#### Example D

*Graph the functions* \begin{align*}y=\sqrt{x}, y = \frac{1}{2} \sqrt{x}, y = \frac{1}{3} \sqrt{x},\end{align*} *and* \begin{align*}y = \frac{1}{4} \sqrt{x}\end{align*} *on the same graph.*

**Solution**

If we multiply the function by a constant between 0 and 1, the function increases more slowly the smaller the constant is.

Watch this video for help with the Examples above.

CK-12 Foundation: Graphs of Square Root Functions

### Guided Practice

*Graph the functions*

a) \begin{align*}y=2 \sqrt{x}\end{align*} *and* \begin{align*}y=-2 \sqrt{x}\end{align*} *on the same graph.*

b) \begin{align*}y=\sqrt{x}\end{align*} *and* \begin{align*}y = \sqrt{-x}\end{align*} *on the same graph.*

**Solutions:**

a) If we multiply the whole function by -1, the graph is reflected about the \begin{align*}x-\end{align*}axis.

b)

On the other hand, when just the \begin{align*}x\end{align*} is multiplied by -1, the graph is reflected about the \begin{align*}y-\end{align*}axis. Notice that the function \begin{align*}y=\sqrt{-x}\end{align*} has only negative \begin{align*}x-\end{align*}values in its domain, because when \begin{align*}x\end{align*} is negative, the expression under the radical sign is positive.

### Explore More

Graph the following functions.

- \begin{align*}y = 3\sqrt{x}\end{align*}
- \begin{align*}y = -\frac{1}{2}\sqrt{x}\end{align*}
- \begin{align*}y = \sqrt{4x}\end{align*}
- \begin{align*}y = \sqrt{x}+7\end{align*}
- \begin{align*}y = 2\sqrt{x}-5\end{align*}
- \begin{align*}y = -\sqrt{3x+1}-2\end{align*}

Graph the following functions on the same coordinate axes.

- \begin{align*}y = \sqrt{x}, y = 2.5\sqrt{x}\end{align*} and \begin{align*} y= -2.5\sqrt{x}\end{align*}
- \begin{align*}y = \sqrt{x}, y = 0.3 \sqrt{x}\end{align*} and \begin{align*} y= 0.6\sqrt{x}\end{align*}
- \begin{align*}y = \sqrt{x}, y = \sqrt{x - 5}\end{align*} and \begin{align*} y= \sqrt{x + 5}\end{align*}
- \begin{align*}y = \sqrt{x}, y = \sqrt{x} + 8\end{align*} and \begin{align*} y= \sqrt{x} - 8\end{align*}

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 11.1.