Power Functions In Math: Exploring The Influence Of Constants And Variables On Graphs

power function graph

y(x)=(L/1+Ce^-k(x-h))+v

A power function is a function of the form f(x) = ax^k, where a and k are constants and x is a variable. The graph of a power function depends on the value of k, as well as the sign of a.

If k is an even positive integer (e.g. k = 2, 4, 6), the graph of the power function will be a downward or upward opening parabola, depending on the sign of a. Specifically, if a is positive, the graph will open upward and if a is negative, the graph will open downward. Examples of power functions with even positive integer values of k include f(x) = x^2 and f(x) = -x^4.

If k is an odd positive integer (e.g. k = 1, 3, 5), the graph of the power function will be a straight line that passes through the origin. The slope of the line will depend on the sign of a. Specifically, if a is positive, the line will have a positive slope and if a is negative, the line will have a negative slope. Examples of power functions with odd positive integer values of k include f(x) = x and f(x) = -x^3.

If k is negative (e.g. k = -1, -2, -3), the graph of the power function will be a curve that approaches the x-axis as x approaches infinity or negative infinity, depending on the sign of a. Specifically, if a is positive, the curve will approach the x-axis from above and if a is negative, the curve will approach the x-axis from below. Examples of power functions with negative values of k include f(x) = 1/x and f(x) = -1/x^2.

It is important to note that the behavior of the graph near the origin will also depend on the value of a, even when k is positive. For example, the graph of f(x) = x^2 will be a parabola that opens upward for positive values of a and downward for negative values of a, but the graph will also intersect the y-axis at the point (0, a).

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