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You are reading an older version of this FlexBook® textbook: CK-12 Algebra I Teacher's Edition Go to the latest version.

11.1: Graphs of Square Root Functions

Difficulty Level: 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.


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

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 y=\sqrt{x}.

a. y=-\sqrt{x}

Hint: Flip about the x-axis.

b. y=\sqrt{-x}

Hint: Flip about the y-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. y=2\sqrt{x}

Hint: Stretch in the vertical direction by a factor of 2: y-values are multiplied by 2.

d. y=\sqrt{2x}

Hint: Point out that y=\sqrt{2x}=\sqrt{2}\sqrt{x} .

e. y=\sqrt{x}+2

Hint: Shift the graph up by 2: y-values are increased by 2.

f. y=\sqrt{x+2}

Hint: Shift the graph left by 2: x-values are decreased by 2.

g. y=-\sqrt{x}+1

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

h. y=\sqrt{-x+1}

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

Additional Example:

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

a. y=4+2\sqrt{2-x}

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

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

To graph y=2\sqrt{2-x} we multiply the y-values of y=\sqrt{2-x} by 2 to obtain a vertically stretched curve. Finally, to obtain the graph of y=4+2\sqrt{2-x}, shift the graph of y=2\sqrt{2-x} 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: f(x)=mx+b

Examples: f(x)=x, \ f(x)=-x, \ f(x)=x+1, \ f(x)=2

  • Exponential: f(x)=a \cdot b^x

Examples: f(x)=2^x, \ f(x)=2^{-x}, \ f(x)=-2^x

  • Quadratic: f(x)=ax^2+bx+c

Examples: f(x)=x^2, \ f(x)=-x^2, \ f(x)=x^2+1

  • Square root: f(x)=a\sqrt{bx+c}+d

Examples: f(x)=\sqrt{x}, \ f(x)=\sqrt{x+1}, \ f(x)=\sqrt{x}+1, \ f(x)=-\sqrt{x}

Error Troubleshooting

General Tip: Students may not recognize y=\sqrt{-x} 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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Date Created:

Feb 22, 2012

Last Modified:

Aug 22, 2014
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