why does a direct variation graph have to go through the origin?
a direct variation graph has the form
y = kx, where k is a constant
compare this with the equation y = mx + b
if b=0, the line goes through the origin, or (0.0) satisfies both equations.
A direct variation graph, also known as a linear function, has a specific equation form of y = kx, where k is a constant called the constant of variation. The graph of this equation is a straight line that passes through the origin (0,0). Here's why:
1. Definition of Direct Variation: In a direct variation, when one variable increases or decreases, the other variable changes proportionally. This means that as x increases or decreases, y changes by a constant factor.
2. The Constant of Variation: In the equation y = kx, the constant of variation (k) represents this constant factor. It shows how much y changes for every change in x. If k is positive, the line slopes upward from left to right, indicating that y increases as x increases. If k is negative, the line slopes downward, indicating that y decreases as x increases.
3. Relationship at the Origin: Since the origin (0,0) represents the absence of any change in x or y, it follows that if x and y are proportional, they both need to be zero at the origin. In other words, when x is zero, y must also be zero. Therefore, a direct variation graph must pass through the origin.
4. Differentiation from Other Linear Equations: It is important to note that not all linear equations have a direct variation pattern and pass through the origin. For example, the equation y = mx + b represents a general linear equation where b is the y-intercept, meaning the line does not go through the origin.
In summary, a direct variation graph goes through the origin because both variables (x and y) are proportional and have no change when they are zero.
A direct variation graph goes through the origin because the relationship between the two variables is linear and proportional. In a direct variation, as one variable increases, the other variable increases or decreases at a constant rate. This constant rate of change is called the constant of variation.
To understand why a direct variation graph goes through the origin, we can take a look at the equation that represents it. In general, a direct variation can be expressed as:
y = kx
where y and x are the two variables and k is the constant of variation. The equation shows that y is directly proportional to x and k represents the constant rate of change.
When the value of x is 0, it follows that y must also be 0 in order for the equation to hold true. This means that when there is no value for x, there is no value for y as well. This point where both variables have a value of zero is the origin (0, 0) on a graph. Therefore, a direct variation graph must go through the origin.
In mathematical terms, when x = 0, we substitute this value into the equation:
y = k * 0
Since anything multiplied by zero is always zero, we get:
y = 0
This shows that the y-coordinate is always zero when x is zero, ensuring that the graph passes through the origin.