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11.1: Graphs of Square Root Functions

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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.

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; $x < \sqrt{x}$ 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.

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.

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.

Feb 22, 2012

Aug 22, 2014