In a proportional relationship, why must the graph of the line always go through the origin (0,0)?
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A proportional relationship is represented by
y=mx where m is a constant.
If there is a y-intercept, it would be
y=mx+b
in which case the relationship is no longer proportional.
For example, y=x+4
x y
1 5
2 6
3 7
is not proportional, because
1/5≠2/6≠3/7...
Example of a proportional relationship,
y=2x
x y
1 2
2 4
3 6
4 8
...
means
1/2=2/4=3/6=4/8....
In a proportional relationship, the graph of the line always goes through the origin (0,0) because it represents a direct and constant ratio between the two variables.
To understand why the graph of a proportional relationship passes through the origin, imagine the two variables as coordinates on a graph, with the independent variable, often denoted by x, plotted along the x-axis, and the dependent variable, often denoted by y, plotted along the y-axis.
In a proportional relationship, the ratio between the two variables remains constant. This means that for any given value of x, the corresponding value of y can be determined by multiplying x by a constant factor, often referred to as the constant of proportionality.
Mathematically, a proportional relationship can be represented by the equation y = kx, where k is the constant of proportionality.
Now, let's consider what happens when x is equal to zero in this equation. When x is zero, the equation becomes y = k * 0, which simplifies to y = 0.
It becomes clear that when x is zero, y must also be zero for the equation to hold true. This means that the ordered pair (0,0) is always a solution to a proportional relationship.
As a result, the graph of a proportional relationship must pass through the origin (0,0), because it represents the point where both variables are zero and it is a fundamental property of proportional relationships.