How to determine if the graph shows two quantities that vary directly and how to determine the constant of proportionality?
recall that the graph y = kx
has slope = k, the constant of proportionality
To determine if the graph shows two quantities that vary directly, you need to check if the points on the graph form a straight line that passes through the origin (0,0). Here's how you can do it:
1. Look at the graph: Examine the shape of the graph. If it shows a straight line that passes through the origin (0,0), then it indicates a direct variation.
2. Pattern: Observe the relationship between the x-values and y-values. If the ratio of the y-values to the x-values remains constant, it suggests a direct variation. In other words, if increasing or decreasing values of x directly correspond to the same proportional change in y, it indicates direct variation.
To determine the constant of proportionality in a direct variation, follow these steps:
1. Select two points: Choose any two points that lie on the straight line on the graph. Make sure these points have distinct x and y values.
2. Calculate the ratio: Determine the ratio of the y-values to the x-values for the two chosen points. Divide the y-value of one point by its corresponding x-value. Repeat the same process for the other point.
3. Compare ratios: If the graph represents a direct variation, the ratios calculated in step 2 will be equal.
4. Constant of proportionality: The ratio obtained above represents the constant of proportionality. It is denoted by the symbol k, and it indicates the amount by which y changes per unit change in x.