Transformations on the Graph of the Absolute Value Parent Function
Description/Explanation/Highlights
Video Description
In this video we learn the effect on the graph of the absolute value parent function when performing various arithmetic operations on the parent function.
Steps and Key Points to Remember
To perform basic transformations to the graph of the absolute value parent function use the following steps:
- Begin with the graph of the absolute value parent function: Y=|X|.
- It can be helpful to sketch the graph of the parent with dotted lines for reference.
- Look at any numbers added or subtracted inside or outside of the absolute value signs first.
- Move the vertex up by any number added on the outside or down by any number subtracted on the outside such as Y=|X|+3 or Y=|X|−3.
- Move the vertex right by any number subtracted inside the absolute value sign or left by any number added on the inside such as Y=|X−3| or Y=|X+3|.
- Graph a point at the new adjusted vertex.
- If the function is multiplied by a number on the outside of the absolute value sign such as Y=3|X|, go to the right by one unit and up by the multiplier from the new vertex if positive and down by the multiplier if negative.
- Graph a reference point in this position.
- Repeat the process by going left by one and then up or down by the multiplier.
- You may repeat this process from the point just created to better establish the “slope” on each side of the vertex.
- If the function is multipled by a number on the inside of the absolute value sign such as Y=|3X|, the same process applies as with numbers multiplied on the outside.
- If the function is multiplied by numbers on both the inside and outside of the absolute value sign such as Y=2|3X|, multiply the two numbers together to decide how far to move up or down after moving left or right one. NOTE: Only a negative sign on the outside will cause the graph to open down. Negative signs on the inside should be ignored when determining up or down direction.
- After enough reference points right and left of the vertex have been established, connect the points in both directions from the vertex.
- This is the graph of the transformed function and should still make a basic “V” shape like the parent.
Here are some key points to keep in mind when graphing transformations of the absolute value parent function:
- Always start with the Y=|X|, the parent and simplest form of the absolute value function.
- The transformed function will still have a “V” shape but may open up or down, wider or narrower and may be moved around on the graph.
- Always do translations of the vertex up/down and right/left first.
- Only negative numbers multiplied on the outside of the absolute value signs will determine if the graph opens up or down with positive opening up and negative opening down.
- Multiplying by numbers greater than one (ignoring the sign) on the outside are vertical stretches while multiplying by numbers between zero and one are vertical compressions.
- Multiplying by numbers greater than one (ignoring the sign) on the inside are horizontal compressions while multiplying by numbers between zero and one are horizontal stretches.
- The stretch or compression factor is the number being multiplied by (again ignoring the sign) for vertical stretches and compressions. FOR EXAMPLE: Y=3|X| is a vertical stretch of 3 as is Y=−3|X|.
- The stretch or compression factor is the reciprocal of the number being multiplied by for horizontal stretches and compressions. FOR EXAMPLE: The horizontal compression factor of Y=|3X| is ⅓.
- Vertical stretches and horizontal compressions have the same effect on the graph. FOR EXAMPLE: A vertical stretch of 3 is the same as a horizontal compression of ⅓.
Video Highlights
- 00:00 Introduction to absolute value transformations
- 00:15 Y=|X| absolute value parent graph
- 00:30 Y=|X+3| horizontal translation of the absolute value parent example
- 01:26 Y=|X|+3 vertical translation of the absolute value parent example
- 02:11 Y=−|X| reflection of the absolute value parent over the x-axis example
- 02:40 Y=2|X| vertical stretch of the absolute value parent example
- 03:45 Y=½|X| vertical compression of the absolute value parent example
- 04:35 Y=|2X| horizontal compression of the absolute value parent example
- 05:30 Y=|½X| horizontal stretch of the absolute value parent example
- 06:26 Y=−3|X−2|+4 multiple changes to the absolute value parent example
- 09:22 Conclusion and website information