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11.1: Graphs of Square Root Functions

Difficulty Level: At Grade Created by: CK-12
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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.


Terms introduced in this lesson:

increases, decreases

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 \begin{align*}(0,1)\end{align*} are smaller than their square roots; \begin{align*}x < \sqrt{x}\end{align*} for \begin{align*}x\end{align*} in the interval \begin{align*}(0,1)\end{align*} and \begin{align*}x > \sqrt{x}\end{align*} for \begin{align*}x\end{align*} in the interval \begin{align*}(1,\infty)\end{align*}.

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 \begin{align*}y=\sqrt{x}\end{align*}.

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

Hint: Flip about the \begin{align*}x-\end{align*}axis.

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

Hint: Flip about the \begin{align*}y-\end{align*}axis.

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. \begin{align*}y=2\sqrt{x}\end{align*}

Hint: Stretch in the vertical direction by a factor of \begin{align*}2\end{align*}: \begin{align*}y-\end{align*}values are multiplied by \begin{align*}2\end{align*}.

d. \begin{align*}y=\sqrt{2x}\end{align*}

Hint: Point out that \begin{align*}y=\sqrt{2x}=\sqrt{2}\sqrt{x}\end{align*} .

e. \begin{align*}y=\sqrt{x}+2\end{align*}

Hint: Shift the graph up by \begin{align*}2\end{align*}: \begin{align*}y-\end{align*}values are increased by \begin{align*}2\end{align*}.

f. \begin{align*}y=\sqrt{x+2}\end{align*}

Hint: Shift the graph left by \begin{align*}2\end{align*}: \begin{align*}x-\end{align*}values are decreased by \begin{align*}2\end{align*}.

g. \begin{align*}y=-\sqrt{x}+1\end{align*}

Hint: Point out that the transformations are in the same direction. \begin{align*}y-\end{align*}values are reflected across the \begin{align*}x-\end{align*}axis (parallel to the \begin{align*}y-\end{align*}axis) and then shifted vertically (parallel to the \begin{align*}y-\end{align*}axis), in that order. Therefore, the correct sequence is to flip and then shift.

h. \begin{align*}y=\sqrt{-x+1}\end{align*}

Hint: Have students consider what happens to an input \begin{align*}x\end{align*} and do the transformations in the opposite order. Therefore, shift left, then reflect across the \begin{align*}y-\end{align*}axis.

Additional Example:

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

a. \begin{align*}y=4+2\sqrt{2-x}\end{align*}

Solution: View \begin{align*}y=4+2\sqrt{2-x}\end{align*} as a combination of transformations of the basic square root graph \begin{align*}y=\sqrt{x}\end{align*}.

Start with the simpler equation: \begin{align*}y=\sqrt{2-x}\end{align*}. If we follow an input \begin{align*}x\end{align*}, then it first gets multiplied by \begin{align*}-1\end{align*} and then increased by \begin{align*}2\end{align*}. How do the graphs of \begin{align*}y=\sqrt{x}\end{align*} and \begin{align*}y=\sqrt{2-x}\end{align*} compare? \begin{align*}y=\sqrt{2-x}\end{align*} is a shift of \begin{align*}y=\sqrt{x}\end{align*} two units LEFT and then flipped across the \begin{align*}y-\end{align*}axis. Note that the transformations happen in the opposite order in which an input \begin{align*}x\end{align*} gets operated on.

To graph \begin{align*}y=2\sqrt{2-x}\end{align*} we multiply the \begin{align*}y-\end{align*}values of \begin{align*}y=\sqrt{2-x}\end{align*} by \begin{align*}2\end{align*} to obtain a vertically stretched curve. Finally, to obtain the graph of \begin{align*}y=4+2\sqrt{2-x}\end{align*}, shift the graph of \begin{align*}y=2\sqrt{2-x}\end{align*} four units vertically.

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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