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 are smaller than their square roots; for in the interval and for in the interval .
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 .
Hint: Flip about the axis.
Hint: Flip about the 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 : values are multiplied by .
Hint: Point out that .
Hint: Shift the graph up by : values are increased by .
Hint: Shift the graph left by : values are decreased by .
Hint: Point out that the transformations are in the same direction. values are reflected across the axis (parallel to the axis) and then shifted vertically (parallel to the axis), in that order. Therefore, the correct sequence is to flip and then shift.
Hint: Have students consider what happens to an input and do the transformations in the opposite order. Therefore, shift left, then reflect across the axis.
Graph the following function using shifts, flips, and stretches.
Solution: View as a combination of transformations of the basic square root graph .
Start with the simpler equation: . If we follow an input , then it first gets multiplied by and then increased by . How do the graphs of and compare? is a shift of two units LEFT and then flipped across the axis. Note that the transformations happen in the opposite order in which an input gets operated on.
To graph we multiply the values of by to obtain a vertically stretched curve. Finally, to obtain the graph of , shift the graph of 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:
- Square root:
General Tip: Students may not recognize 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.