Translations of the Rational Parent Function

Description/Explanation/Highlights

Video Description

This video will explain how changes to the rational parent function translate the graph of the parent function up, down, right and left.

Steps and Key Points to Remember

To translate the graph of the rational parent left, right, up or down, follow these steps: Please note: It will be very beneficial to watch the video to see the actual effect on the movement of 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. Any number added or subtracted outside of the fraction will move the graph up or down respectively. For example, \(y=\frac{\textstyle 1}{\textstyle x}-2\) will move the parent graph down 2 and subsequently the horizontal asymptote will also move down 2 to y = -2. If the number had been +2 instead of -2, the graph and asymptote of the graph would have moved up 2. The “shape” of the graph does not change.
  5. Any number that is added or subtracted in the denominator of the fraction moves the graph left or right respectively. In the example \(y=\frac{\textstyle 1}{\textstyle (x+2)}\), the graph of the parent function moves left 2 units and the vertical asymptote is now at -2. If the denominator had been \((x-2)\), the graph and vertical asymptote would have moved to the right 2 units.
  6. Right and left translations can be combined with up and down translations by adding or subtracting numbers in the denominator of the fractions and outside the fraction. For example, \(y=\frac{\textstyle 1}{\textstyle (x-1)}-1\) will move the graph one unit right and one unit down. There will be a new vertical asymptote at x = 1 and a new horizontal asymptote at y = -1.

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

  • The graph of the rational parent function has a vertical asymptote at the y-axis and a horizontal asymptote at the x-axis. This means the graph will approach the x- and y-axis but never reach it.
  • Be careful when using graphing calculators as the graph may appear to disappear or stop at an axis.
  • The graph of the rational parent appears only in quadrants I & III
  • A number added outside of the fraction will move the graph and the horizontal asymptote of the rational parent function up by the number of units equal to the number added.
  • A number subtracted outside of the fraction will have the same effect only in the downward direction.
  • A number added in the denominator of the fraction will move the parent graph and the asymptote left by that number of units.
  • A number subtracted in the denominator will move the graph of the parent and the asymptote to the right by a corresponding number of units.
  • Neither of these translations change the shape of the graph.
  • These operation may be combined to move the graph up or down and right or left at the same time.
  • When graphing, it is often helpful to move the asymptote(s) first and draw them as dotted reference lines.

Video Highlights

  • 00:00 Introduction
  • 00:10 \(y=\frac{\textstyle 1}{\textstyle x}\) rational parent graph
  • 02:53 \(y=\frac{\textstyle 1}{\textstyle x}-2\) example of a vertical translation of the rational parent.
  • 04:46 \(y=\frac{\textstyle 1}{\textstyle (x+2)}\) example of a horizontal translation of the rational parent.
  • 06:29 \(y=\frac{\textstyle 1}{\textstyle (x-1)}-1\) example of a combined translation of the rational parent.
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Dividing Rational Expressions
Solving Rational Equations
Solving More Complex Rational Equations
Reflections and Vertical Stretches of the Rational Parent Function
Vertical Compressions and Multiple Transformations of the Rational Parent
Finding Vertical Asymptotes
Finding Horizontal Asymptotes
Finding Removable Discontinuities
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