Transformations of the Square Root Parent Function (Translations & Reflections)
Description/Explanation/Highlights
Video Description
This video explains how to do translations and reflections 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 translations and reflections, 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 moved around are: (0, 0), (1, 1), (4, 2), and (9, 3).
- When a number is added or subtracted on the “outside” of the square root sign, the graph and key points are moved up (+) or down (-) by that number.
- In the example, \(y=\sqrt{x}+3\), the graph and key points will be moved up three units. This has the effect of adding 3 to the y-value of each coordinate. The “shape” of the graph is unchanged. New key points are (0, 3), (1, 4), (4, 5), and (9, 6).
- Subtracting 3 on the “outside” would have had the opposite effect and moved the graph down 3 units. Y-values would have 3 subtracted to get new key points.
- Numbers added or subtracted under the square root or “inside” would have the effect of moving the graph right (-) or left (+). Values would be subtracted or added to the x-values of the coordinates of the key points. Note: values added under the square root are subtracted from the x-values and numbers subtracted under the square root are added to the x-values.
- \(y=\sqrt{x+3}\) would move the parent function left by three units. Subtracting 3 would have moved the graph right by a corresponding number of units.
- Adding a negative sign in front of the square root will cause the graph to reflect over the x-axis and each y-value of the coordinates to be negative in the key points.
- \(y=-\sqrt{x}\) would reflect the parent graph over the x-axis and new key points would be: (0, 0), (1, -1), (4, -2), and (9, -3).
- Putting the negative “inside” the square root would cause the graph to reflect over the y-axis and make the x-value of the coordinates negative.
- \(y=\sqrt{-x}\) will reflect over the y-axis.
- These transformations can be combined. For example, \(y=-\sqrt{x+2}-3\), will move the parent down 3, left 2, and reflect it over the x-axis while \(y=\sqrt{-(x+2)}-3\) will move the graph down 3, left 2, and reflect it over the y-axis.
- \(y=-\sqrt{x+7}+3\) will move the parent graph left 7, up 3, and reflect it over the x-axis.
Here are some key points to keep in mind when translating and reflecting graphs 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.
- Adding or subtracting a constant on the “outside” of the square root moves the graph of the parent function up or down respectively.
- The constant is added or subtracted from the y-values of the coordinates of key point.
- Constants added or subtracted “inside” or under the square root sign move the graph left or right respectively and subtract or add the constant from the x-value of the coordinates of the key points.
- Be careful when adjusting the key points for left and right translations. When a number is subtracted under the square root sign, it must be ADDED to the x-values and when a number is added under the square root, it must be SUBTRACTED from the x-values to move the graph in the correct direction.
- A negative sign in front of the square root will reflect the graph over the x-axis.
- A negative sign in front of x under the square root will reflect the graph over the y-axis.
- It is generally easier when several transformations occur at once, to apply the up/down and left/right moves first followed by any reflection over the x- or y-axis.
Video Highlights
- 00:00 Introduction
- 00:33 \(y=\sqrt{x}\) parent function
- 01:10 \(y=\sqrt{x}+3\) example of a vertical translation
- 02:17 \(y=\sqrt{x+3}\) example of a horizontal translation
- 03:22 \(y=-\sqrt{x}\) example of a reflection over the x-axis
- 04:08 \(y=\sqrt{-x}\) example of a reflection over the y-axis
- 04:45 \(y=-\sqrt{x+2}-3\) example of a multiple transformation
- 06:00 \(y=\sqrt{-(x+2)}-3\) example of a multiple transformation
- 07:50 Example of writing an equation from a graph by applying transformations to find \(y=-\sqrt{x+7}+3\)
- 09:30 Conclusion
- To watch this video on YouTube in a new window with clickable highlights, click here