The power function equation y = kx^a represents a relationship between two variables, x and y, where y is proportional to some power of x. The constant k in the equation determines the overall scale of the function and affects how the graph of the function looks.
1. When k increases, the graph of the function stretches vertically. In other words, the entire graph is multiplied by a constant factor. This means that for any given value of x, the corresponding value of y will be larger than it would be with a smaller value of k.
2. When k decreases, the graph of the function shrinks vertically. In other words, the entire graph is divided by a constant factor. This means that for any given value of x, the corresponding value of y will be smaller than it would be with a larger value of k.
3. When k becomes negative, the graph of the function reflects across the x-axis. In other words, the entire graph is flipped over the x-axis. This means that for any given value of x, the corresponding value of y will have the opposite sign as it would have with a positive value of k. For example, if k is positive and the graph is above the x-axis, then if k is negative the graph will be below the x-axis.
It's worth noting that changing the value of k does not affect the shape of the graph. The shape of the graph is determined solely by the value of the exponent a. However, the value of k does affect how the graph is scaled and where it is positioned relative to the x-axis and y-axis.