# 11.1: Graphs of Square Root Functions

**At Grade**Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

- Graph and compare square root functions.
- Shift graphs of square root functions.
- Graph square root functions using a graphing calculator.
- Solve real-world problems using square root functions.

## Vocabulary

Terms introduced in this lesson:

- increases, decreases
- flip
- shift
- stretch
- transform

## Teaching Strategies and Tips

Use Example 1 to introduce the basic shape of the square root function.

Use the tables in the examples to show students the square root function’s behavior numerically:

- Why it is undefined on some intervals.
- That it is everywhere increasing.
- That it rises relatively slowly.
- That the square root of a fraction is greater than the fraction. Numbers in the interval
(0,1) are smaller than their square roots;x<x√ forx in the interval(0,1) andx>x√ forx in the interval(1,∞) .

Use the graphs in the examples to make the following observations:

- The square root graph is half a parabola lying sideways.

Emphasize finding the domain of the square root function before making a table.

- When the expression under the square root is negative, table values will be undefined; and the graph corresponding to the interval will be empty.

Use Examples 2-10 to motivate transformations:

- Shifts, stretches, and flips allow graphing
*without*constructing a table of values. - Teachers are encouraged to use several examples to illustrate the effect of each constant on the graph.

Additional Examples:

*Graph the following functions using transformations of the basic graph*

a.

Hint: Flip about the

b.

Hint: Flip about the

Remark: Student often claim that the whole function is undefined because of the negative under the radical. Point out that the domain “flips” to negative numbers.

c.

Hint: Stretch in the vertical direction by a factor of

d.

Hint: Point out that

e.

Hint: Shift the graph up by

f.

Hint: Shift the graph left by

g.

Hint: Point out that the transformations are in the same direction.

h.

Hint: Have students consider what happens to an input

Additional Example:

*Graph the following function using shifts, flips, and stretches.*

a.

Solution: View

Start with the simpler equation:

To graph

Encourage students to keep a list of functions they have studied so far. Include a few examples of each and their graphs. For example:

- Linear: \begin{align*}f(x)=mx+b\end{align*}

Examples: \begin{align*}f(x)=x, \ f(x)=-x, \ f(x)=x+1, \ f(x)=2\end{align*}

- Exponential: \begin{align*}f(x)=a \cdot b^x\end{align*}

Examples: \begin{align*}f(x)=2^x, \ f(x)=2^{-x}, \ f(x)=-2^x\end{align*}

- Quadratic: \begin{align*}f(x)=ax^2+bx+c\end{align*}

Examples: \begin{align*}f(x)=x^2, \ f(x)=-x^2, \ f(x)=x^2+1\end{align*}

- Square root: \begin{align*}f(x)=a\sqrt{bx+c}+d\end{align*}

Examples: \begin{align*}f(x)=\sqrt{x}, \ f(x)=\sqrt{x+1}, \ f(x)=\sqrt{x}+1, \ f(x)=-\sqrt{x}\end{align*}

## Error Troubleshooting

General Tip: Students may not recognize \begin{align*}y=\sqrt{-x}\end{align*} as a valid function at first, stating that the square root of a negative is undefined. Explain that the function’s domain is defined.

General Tip: Have students find the domain of the function they are graphing on a calculator first. This will help find an appropriate window for the graph.

In Example 12 and *Review Questions* 14-18, state beforehand the number of decimal places required of students when rounding.

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