Louis says that if the differences between the values of
x
are constant between all the points on a graph,
then the relationship is linear. Do you agree? Explain.
Yes, I agree with Louis. If the differences between the values of x are constant between all the points on a graph, then the relationship is linear.
To understand why, we need to discuss what it means for a relationship to be linear. In mathematics, a linear relationship between two variables, let's say x and y, can be expressed as y = mx + b, where m represents the slope (rate of change) and b represents the y-intercept (the value of y when x = 0).
Now, let's consider a graph where the differences between the values of x are constant between all the points. This means that as we move from one point to the next, the change in x remains the same. In other words, the x-values are equally spaced along the x-axis.
If we have such a graph, we can calculate the slope between any two points by finding the change in y divided by the change in x. Since the differences between the x-values are constant, the change in x will always be the same. Consequently, the slope between any two points on the graph will be constant.
A key characteristic of linear relationships is that the slope remains constant throughout the entire graph. Therefore, if the differences between the values of x are constant, we can conclude that the relationship is linear.
To summarize, if the differences between the values of x are constant between all the points on a graph, it implies a constant slope, which is a defining characteristic of a linear relationship.