Vertical Compressions and Multiple Transformations of the Rational Parent Function

Description/Explanation/Highlights

Video Description

This video will explain how multiplying by a fraction (vertical compressions) and multiple changes to the rational parent function affect the graph of the parent function.

Steps and Key Points to Remember

To vertically compress or perform multiple transformations to the graph of the rational parent, follow these steps: Please note: It will be very beneficial to watch the video to see the actual effect on the graph for each change to the parent function.

  1. Remember, the rational parent function is identified as \(y=\frac{\textstyle 1}{\textstyle x}\) or \(f(x)=\frac{\textstyle 1}{\textstyle x}\).
  2. The graph of the rational parent is unique in that it appears only in quadrants I & III. As the value of x approaches negative infinity, the graph approaches the x-axis from below and as the value approaches x = 0, the graph turns downward at (-1, -1) and approaches the y-axis.  As the value of x approaches x = 0 from the right, the graph turns upward at (1, 1) and approaches the y-axis from the right. As the value of x approaches infinity, the graph approaches the x-axis from above.
  3. Since the graph only “approaches” the x- and y-axis but never reaches them, horizontal and vertical asymptotes are created at the x- and y-axis. This is important because as the graph is translated, these asymptotes will move with the translations.
  4. A vertical compression occurs when the rational parent is multiplied by a fraction or the variable in the denominator is multiplied by a number greater than 1.
  5. For example \(y=\frac{\textstyle 1}{\textstyle 2}(\frac{\textstyle 1}{\textstyle x})\) or as it can also be written, \(y=\frac{\textstyle 1}{\textstyle 2x}\) is a vertical compression by a factor of \(\frac{\textstyle 1}{\textstyle 2}\).
  6. Just as with vertical stretches, multiply the y-coordinates of the “turning points” of the parent graph [(-1, -1) & (1, 1)] in quadrant I & III by the compression factor.
  7. In the above example, the new coordinates will be \((-1, -\frac{\textstyle 1}{\textstyle 2})\) and \((1, \frac{\textstyle 1}{\textstyle 2})\).
  8. This has the effect of pushing down the “turning point” of the graph and compressing the graph.
  9. Any rational function can be viewed as a multiple transformation to the parent when graphing it.
  10. Multiple transformations involving only vertical and horizontal translations are best accomplished by first locating the new horizontal and vertical asymptotes and drawing them as dotted reference lines.
  11. In the example \(y=\frac{\textstyle 1}{\textstyle (x+2)}-1\), the graph will be moved down 1 unit and left 2 units. Locate the new horizontal asymptote by drawing a new horizontal dotted line at \(y=-1\) to represent the vertical translation and a new vertical dotted line at \(x=-2\) to represent the horizontal translation.
  12. Draw the new graph with the new asymptotes as a reference that is the same shape as the parent.
  13. Also note that the “turning points” will have moved down 1 unit to (-1, -2) & (1, 0) and then left 2 units to (-3, -2) and (-1, 0) to their final translated locations.
  14. The example \(y=-\frac{\textstyle 2}{\textstyle x}\), is an example of a vertical stretch of 2 and a reflection over the x-axis when graphed.

Here are some key points to keep in mind when graphing multiple transformations of the rational parent function.

  • A vertical compression graphs like a vertical stretch but the compression factor is between 0 and 1.
  • A vertical compression, graphs like a stretch, but will change the shape of the graph by making the curve “deeper” instead of shallower.
  • More complicated rational functions can be graphed by applying multiple transformations to the graph.
  • Start multiple transformations by doing reflections, and then stretches or compressions.
  • After other transformations are complete, locate the new vertical and horizontal asymptotes, and the coordinates of the “turning points” to translate the graph up/down and left/right.

Video Highlights

  • 00:00 Introduction
  • 00:22 \(y=\frac{\textstyle 1}{\textstyle x}\) rational parent graph and review of translations, vertical stretches, and reflections over the x-axis
  • 01:17 \(y=\frac{\textstyle 1}{\textstyle 2}(\frac{\textstyle 1}{\textstyle x})\) example of a vertical compression of the rational parent by a factor of \(\frac{\textstyle 1}{\textstyle 2}\)
  • 03:18 \(y=\frac{\textstyle 1}{\textstyle (x+2)}-1\) example of a vertical translation combined with a horizontal translation of the rational parent
  • 06:00 \(y=-\frac{\textstyle 2}{\textstyle x}\) example of a vertical stretch combined with a reflection over the x-axis
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Solving Rational Equations
Solving More Complex Rational Equations
Translations of the Rational Parent Function
Reflections and Vertical Stretches of the Rational Parent Function
Finding Vertical Asymptotes
Finding Horizontal Asymptotes
Finding Removable Discontinuities
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