Inverse Variation
Description/Explanation/Highlights
Video Description
This video explains how to recognize and use inverse variation.
Steps and Key Points to Remember
To identify and use inverse variation, follow these steps:
- Inverse variation occurs when y is related to x by a constant product of \(x\cdot y\).
- Equations representing inverse variation take the form of \(y=\frac{\textstyle k}{\textstyle x}\) where \(k\) is the constant of variation. This relationship is an inverse relationship.
- \(y=\frac{\textstyle 200}{\textstyle x}\) is an example of inverse variation. It takes the form of \(y=\frac{\textstyle k}{\textstyle x}\) where \(k=200\). \(k\) is the constant of variation.
- This equation, like all inverse variations, creates a rational function with x in the denominator. This means that the y-values will approach 0 but never equal 0.
- The value of \(k=200\) above could represent, for example, the total cost of renting a condo for one night at $200 per night. If \(y\) represents the cost person, \(y\) is related to \(x\) by the nightly condo total cost with x representing the number of people sharing the cost.
- We can always find the constant of variation by finding the product of \(k=x\cdot y\) for any non-zero value of x and y.
- To find out if the table of values below represents inverse variation, find the value of \(k\) for each x and y value in the table. If all the \(k\) values are the same, it is a inverse variation.
X
2
4
8
Y
100
50
25
- We need to see if the value of \(k\) is the same for each x, y coordinate.
- \(k=1\cdot 200=200\)
- \(k=2\cdot 100=200\)
- \(k=4\cdot 50=200\)
- \(k=8\cdot 25=200\)
- Since all of the x, y coordinates result in \(k=200\), the table represents inverse variation.
- The equation for the inverse variation represented in the table is: \(y=\frac{\textstyle k}{\textstyle x}=\frac{\textstyle 200}{\textstyle x}\) so the equation is \(y=\frac{\textstyle 200}{\textstyle x}\).
- If we know that \(y\) varies inversely with \(x\) and we know the values of one of the x, y pairs, we can find the value of \(y\) for any value of \(x\) by finding the constant of variation and then the equation of the inverse variation.
- For example, if we are given that \(y\) varies inversely with \(x\) and the value of \(y=20\) when \(x=5\), how can we find \(y\) when \(x=10\)?
- Find k from the given x and y values. \(k=x\cdot y=5\cdot 20=100\)
- Find the equation of the inverse variation and substitute the new value for x. \(y=\frac{\textstyle k}{\textstyle x}\longrightarrow y=\frac{\textstyle 100}{\textstyle x}\longrightarrow y=\frac{\textstyle 100}{\textstyle 10}\longrightarrow y=10\)
Here are some key points to keep in mind when identifying and using inverse variations.
- Inverse variation is identified by a rational equation in the form of \(y=\frac{\textstyle k}{\textstyle x}\) where the value of \(k\) represents a constant product.
- To find the value of \(k\), multiply any value of \(y\) by its corresponding x-value. \(k=x\cdot y\).
- Values of x will increase as values of y decrease in inverse variations. Values of x will decrease as values of y increase in inverse variations.
- In an inverse variation all x, y values in a table will have the same value for \(k\) when the product is calculated for \(k=x\cdot y\).
- To find the equation for an inverse variation, calculate \(k\) for one x, y pair and substitute it for \(k\) in the equation \(y=\frac{\textstyle k}{\textstyle x}\).
Video Highlights
- 00:00 Introduction
- 00:05 Definition of inverse variation and the constant \(k\)
- 00:45 \(y=\frac{\textstyle 200}{\textstyle x}\) example of inverse variation
- 02:25 Example of identifying and writing the equation of an inverse variation from a table
- 03:50 Example of finding a missing variable in an inverse variation from a word problem
- To watch this video on YouTube in a new window with clickable highlights, click here