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:
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√ for x in the interval (0,1) and x>x√ for x 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.
Graph the following functions using transformations of the basic graph y=x√.
Hint: Flip about the x−axis.
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.
Hint: Stretch in the vertical direction by a factor of 2: y−values are multiplied by 2.
Hint: Point out that y=2x−−√=2√x√ .
Hint: Shift the graph up by 2: y−values are increased by 2.
Hint: Shift the graph left by 2: x−values are decreased by 2.
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.
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.
Solution: View y=4+22−x−−−−−√ as a combination of transformations of the basic square root graph y=x√.
Start with the simpler equation: y=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=x√ and y=2−x−−−−−√ compare? y=2−x−−−−−√ is a shift of y=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=22−x−−−−−√ we multiply the y−values of y=2−x−−−−−√ by 2 to obtain a vertically stretched curve. Finally, to obtain the graph of y=4+22−x−−−−−√, shift the graph of y=22−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:
Examples: f(x)=x, f(x)=−x, f(x)=x+1, f(x)=2
Examples: f(x)=2x, f(x)=2−x, f(x)=−2x
Examples: f(x)=x2, f(x)=−x2, f(x)=x2+1
- Square root: f(x)=abx+c−−−−−√+d
Examples: f(x)=x√, f(x)=x+1−−−−−√, f(x)=x√+1, f(x)=−x√
General Tip: Students may not recognize y=−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.