The graph of a power function is
f(x)=kx^p, where k and p are constants. Do you expect k to be positive or negative ? Explain please.
Power functions usually are encountered in things like population or income investment growth. In such situations, the k would be positive. There is no reason why k should not be negative, in general.
To determine whether k should be positive or negative in the power function f(x) = kx^p, we can break it down by analyzing the behavior of the graph of the power function.
First, let's consider the behavior of the function when x is positive. We notice that if p is an even number (such as 2, 4, 6, etc.), then regardless of the value of k, the graph will always be positive. This means that k could be either positive or negative, and the graph will still lie entirely above the x-axis.
However, if p is an odd number (such as 1, 3, 5, etc.), the behavior of the graph changes. When x is positive, the value of f(x) will be positive if k is positive. On the other hand, if k is negative, the graph will be negative when x is positive.
Therefore, based on the behavior of the function, we can conclude that when p is an odd number, k should be positive for the graph to lie entirely above the x-axis. When p is an even number, k can be either positive or negative, as the graph will still be above the x-axis.
Note: These observations are based on the assumption that k is not equal to zero. If k is equal to zero, the graph will be a straight line at the x-axis.