Direct Variatiion
Description/Explanation/Highlights
Video Description
This video explains how to recognize and use direct variation.
Steps and Key Points to Remember
To identify and use direct variation, follow these steps:
- Direct variation occurs when y is related to x by a constant ratio or rate.
- Equations representing direct variation take the form of \(y=kx\) where \(k\) is the constant of variation.
- \(y=3x\) is an example of direct variation. It takes the form of \(y=kx\) where \(k=3\). \(k\) is the constant rate or constant of variation.
- Notice that the equation always has a y-intercept of \((0,0)\) when graphed in a direct variation.
- The value of \(k=3\) above could represent, for example, the cost of an item on sale for $3. If \(y\) represents the total cost of a purchase, \(y\) is related to \(x\) by the constant of $3 per item with x representing the number of items purchased. Notice that it is necessary for 0 items purchased to cost $0 for the relationship to work.
- We can always find the constant of variation by finding the ratio of \(k=\frac{\textstyle y}{\textstyle x}\) for any non-zero value of x and y.
- To find out if the table of values below represents direct variation, find the value of \(k\) for each x and y value in the table. If all the \(k\) values are the same and the y-intercept is \((0,0)\), it is a direct variation.
X
2
4
6
8
Y
6
12
18
24
- Notice that the y-intercept is at \((0,0)\) so we need to see if the value of \(k\) is the same for each x, y coordinate.
- \(k=\frac{6}{2}=3\)
- \(k=\frac{12}{4}=3\)
- \(k=\frac{18}{6}=3\)
- \(k=\frac{24}{8}=3\)
- Since all of the x, y coordinates result in \(k=3\) and the y-intercept is at \((0,0)\), the table represents direct variation.
- The equation for the direct variation represented in the table is: \(y=kx=3x\) so the equation is \(y=3x\).
- If we know that \(y\) varies directly 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 direct variation.
- For example, if we are given that \(y\) varies directly with \(x\) and the value of \(y=10\) when \(x=5\), how can we find \(y\) when \(x=11\)?
- Find k from the given x and y values. \(k=\frac{y}{x}=\frac{10}{5}=2\)
- Find the equation of the direct variation and substitute the new value for x. \(y=kx\longrightarrow y=2x\longrightarrow y=2(11)\longrightarrow y=22\)
Here are some key points to keep in mind when identifying and using direct variations.
- Direct variation is identified by a linear equation (\(y=mx+b\)) in the form of \(y=kx\) where the value of \(b\) is 0 and \(k\) represents a constant ratio or rate.
- To find the value of \(k\), divide any value of \(y\) by its corresponding x-value. \(k=\frac{y}{x}\).
- All direct variations have a y-intercept at \((0,0)\).
- In a direct variation all x, y values in a table will have the same value for \(k\) when the ratio is calculated for \(k=\frac{y}{x}\).
- To find the equation for a direct variation, calculate \(k\) for one x, y pair and substitute it for \(k\) in the equation \(y=kx\).
Video Highlights
- 00:00 Introduction
- 00:05 Definition of direct variation and the constant \(k\)
- 00:30 \(y=3x\) example of direct variation
- 01:25 Example of identifying and writing the equation of a direct variation from a table
- 03:25 Example of finding a missing variable in a direct variation from a word problem
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