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# Power Equations and their Graphs

To represent patterns that are more complicated than straight lines, a different type of equation is needed. Power functions are capable of representing patterns in curved graphs. The equation of a power function always follows the pattern:

y = k Â· x p,

where:

â€¢ The letter x represents the values of the input,

â€¢ The letter y represents the values of the output,

â€¢ k is a number called the constant of proportionality, and,

â€¢ p is a number called the power.

## The Graph of a Power Equation

Power equations are excellent tools for representing three basic types of pattern. The appearance of the graph generated by the power equation depends very strongly on the numerical value of the power, p, as shown in Figure 1 (below)

Figure 1: When the constant of proportionality, k, is positive the appearance of the graph generated by a power equation depends on the numerical value of the power, p. When p is negative (p < 0) the graph decreases, flattening out as it goes. When p is between 0 and 1 (0 < p < 1) the graph increases, flattening out as it goes. When p is greater than 1 (p > 1) the graph increases, getting steeper and steeper as it goes.

The relationship between the numerical value of the power p in the power equation:

y = k Â· x p,

and the appearance of the graph generated by the power equation are summarized in Table 1 (below).

 Value of the power, p Value of the power, p (Symbols) Appearance of the graph generated by the power equation Negative p < 0 When read from left to right the graph decreases in height, getting flatter and flatter as it goes. Between zero and one 0 < p < 1 When read from left to right the graph increases in height, getting flatter and flatter as it goes. Greater than one p > 1 When read from left to right the graph increases in height, getting steeper and steeper as it goes.

Table 1: Summary of the relationship between the power, p, and the appearance of the graph.