Reflections and Vertical Stretches of the Rational Parent Function

Description/Explanation/Highlights

Video Description

This video will explain how changes to the rational parent function reflect and vertically stretch the graph of the parent function.

Steps and Key Points to Remember

To reflect and vertically stretch 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. 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 reflection of the parent occurs when a negative sign is placed in front of the function.
  5. \(y=-\frac{\textstyle 1}{\textstyle x}\) will reflect the graph of the parent over the x-axis or the horizontal asymptote if it has been translated. The portion in quadrant III will be reflected into quadrant II and the portion in quadrant I will reflect to quadrant IV.
  6. The shape will be the same but as a mirror image in the new quadrants after reflection.
  7. A vertical stretch occurs when the curve of the graph as it changes from approaching the horizontal asymptote to approaching the vertical asymptote becomes more “shallow”. This will change the point where the “turn” takes place from (-1, -1) in quadrant III and (1, 1) in quadrant I to a new coordinate based on the value of the numerator. It has the effect of “pulling” the point down/up and stretching it.
  8. In the example, \(y=\frac{\textstyle 2}{\textstyle x}\), the 2 in the numerator causes the y-values of -1 and 1 in the turning points of the parent to be multiplied by 2. The new points where the graph turns are (-1, -2) and (1, 2).
  9. If you will be doing translations as well as a stretch, do the stretch first to establish the point and move the point with the translation relative to the new asymptote.

Here are some key points to keep in mind when graphing reflections or vertical stretches of the rational parent function.

  • When a negative (-) sign is added in front of the rational parent function, the graph of the function is reflected over the x-axis.
  • This reflection will cause the part of the graph in quadrant III to reflect a mirror image in quadrant II and the part in quadrant III will no longer exist.
  • The same thing will happen in quadrant I. A mirror image of quadrant I will appear in quadrant II replacing the graph in quadrant I.
  • Increasing the value of the numerator to a number larger than 1 will cause a vertical stretch. 
  • Multiply the y-values of -1 and 1 in the turning coordinate of the parent function [ (-1, -1) & (1, 1) ] by the number in the numerator.

Video Highlights

  • 00:00 Introduction
  • 00:05 \(y=\frac{\textstyle 1}{\textstyle x}\) rational parent graph and review of translations
  • 01:00 \(y=-\frac{\textstyle 1}{\textstyle x}\) example of a reflection of the rational parent over the x-axis.
  • 02:05 \(y=\frac{\textstyle 2}{\textstyle x}\) example of a vertical stretch of the rational parent.
  • 03:58 Conclusion
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  • To watch this video on YouTube in a new window with clickable highlights, click here

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