why is y=x^2 not a linear function
Find y for various values of x to see why.
The function y = x^2 is not a linear function because it does not have a constant rate of change. In a linear function, the graph is a straight line, and the change in y for a given change in x is always the same (constant). However, in the function y = x^2, as x increases or decreases, the corresponding values of y change at an increasing rate. The graph of y = x^2 is a curved line (a parabola), not a straight line, indicating a non-linear relationship between x and y.
To understand why y = x^2 is not a linear function, we have to understand what it means for a function to be linear.
A linear function is characterized by a constant rate of change, which implies that the graph of the function forms a straight line. In other words, if we increase or decrease the input (x) by a certain amount, the output (y) will increase or decrease by a consistent amount.
Now, let's analyze the function y = x^2. If we choose two points on the graph, say (1,1) and (2,4), and calculate the rate of change between these points, we can see that it is not constant.
For example, when x changes from 1 to 2, y changes from 1 to 4. So, the rate of change is (4-1)/(2-1) = 3/1 = 3. However, if we choose another pair of points, say (2,4) and (3,9), the rate of change is (9-4)/(3-2) = 5/1 = 5.
As you can see, the rate of change is different for different intervals of x. This indicates that the graph of y = x^2 is not a straight line, and therefore, it is not a linear function.