Transformations of the Square Root Parent Function (Stretches and Compressions)
Description/Explanation/Highlights
Video Description
This video explains how to do vertical and horizontal stretches and compressions to the square root parent function. Please Note: It will be beneficial to watch the video to see the actual movements on the graph as a result of changes to the parent function.
Steps and Key Points to Remember
To graph changes to the square root parent function that result in stretches and compressions, follow these steps:
- The graph of the square root parent function begins at point (0, 0) and is drawn only in quadrant I since both the domain and range of the square root parent are both greater than or equal to zero.
- Some of the key points on the graph of the parent function that are good to know as the graph is transformed are: (0, 0), (1, 1), (4, 2), and (9, 3).
- A vertical stretch or compression of the square root parent function occurs when the parent function is multiplied by a number in front of the square root. The number it is multiplied by is called the factor. If the factor is greater than 1, a vertical stretch will occur and if the number is between 0 and 1, a vertical compression occurs.
- The effect of this transformation is to change the shape or curve of the graph by “stretching” it upward or “compressing” it downward.
- In the example, \(y=2\sqrt{x}\), the parent function is stretched upward by a factor of 2. Y-values of the coordinates of the key points are multiplied by the factor of 2 giving new points of (0, 0), (1, 2), (4, 4), and (9, 6) changing the curve of the graph upward from the starting point of (0, 0).
- In the example, \(y=\frac{\textstyle 1}{2}\sqrt{x}\), the parent function is compressed vertically by a factor of \(\frac{\textstyle 1}{2}\) making the graph flatter. Each y-value is multiplied by the factor giving new points of (0, 0), (1, 0.5), (4, 1), and (9, 1.5).
- Horizontal stretches and compressions occur when the function is multiplied by a constant under the square root sign. They work a whole lot like the vertical ones but affect the x-values AND THE FACTOR IS DETERMINED BY APPLYING THE RECIPROCAL of the constant that the function is multiplied by. Because of this, numbers greater than 1 create compressions while numbers between 0 and 1 create stretches.
- In the example \(y=\sqrt{\frac{\textstyle 1}{2}x}\), the reciprocal of \(\frac{\textstyle 1}{2}\) is 2 which creates a horizontal stretch of 2. All of the x-values in the key point coordinates are multiplied by 2 “stretching” the graph horizontally.
- In the example, \(y=\sqrt{2x}\), the reciprocal of 2 is \(\frac{\textstyle 1}{2}\) creating a horizontal compression of \(\frac{\textstyle 1}{2}\). Each x-value is multiplied by the factor to create a horizontal compression of the original graph.
Here are some key points to keep in mind when transforming with stretches and compressions to the graph of the square root parent function.
- The graph of the square root parent function begins at point (0, 0) and is drawn only in quadrant I since the domain and range of the square root parent function are both greater than or equal to zero.
- Multiplying the parent by a number greater than 1 on the outside of the square root creates a vertical stretch.
- Multiplying the parent by a number between 0 and 1 on the outside of the square root creates a vertical compression.
- To find new key points, multiply the y-values of the parent function key points by the stretch or compression factor.
- Horizontal stretches are created by multiplying the function under the square root sign by a number between 0 and 1.
- Multiplying by a number under the square root sign greater than 1 creates a horizontal compression.
- To find new key points in a horizontal stretch or compression, multiply the x-values by a compression factor created by using the RECIPROCAL of the number multiplied under the square root sign.
- The effect of applying a vertical stretch is similar to applying a horizontal compression and the effect of applying a vertical compression is similar to applying a horizontal stretch.
Video Highlights
- 00:00 Introduction
- 00:27 \(y=\sqrt{x}\) parent function review
- 01:05 \(y=2\sqrt{x}\) example of a vertical stretch
- 02:30 \(y=\frac{\textstyle 1}{2}\sqrt{x}\) example of a vertical compression
- 03:25 \(y=\sqrt{2x}\) example of a horizontal compression
- 04:53 \(y=\sqrt{\frac{\textstyle 1}{2}x}\) example of a horizontal stretch
- 06:05 \(y=-2\sqrt{x+2}-3\) example of a multiple transformation including a vertical stretch
- 08:10 Conclusion
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