In mathematics, a function is a relation between sets that assigns to each element of a first set, exactly one element of the second set. A graph of a function is a visual representation of the relationship between the two sets. In order to complete an equation for a function graphed above, one must first identify the type of function. The type of function is often determined by the form of the graph.

## Types of Functions

The three most common types of functions are linear, quadratic, and exponential. Linear functions are graphed as a straight line, and can be described by a simple equation of the form y=mx+b, where m is the slope of the line and b is the y-intercept. Quadratic functions are graphed as a parabola, and can be described by a more complex equation of the form y=ax^{2}+bx+c, where a, b, and c are constants. Exponential functions are graphed as an exponential curve, and can be described by an equation of the form y=a^{x}+b, where a and b are constants.

## Finding the Equation

Once the type of function is determined, one can complete the equation by analyzing the graph. For linear functions, the slope and y-intercept can be determined by inspecting the graph. The equation can then be written as y=mx+b, where m and b are the determined values. For quadratic and exponential functions, the constants can be determined by finding the x- and y-intercepts of the graph. The equation can then be written as y=ax^{2}+bx+c for quadratic functions and y=a^{x}+b for exponential functions, where a, b, and c are the determined values.

## Examples

For example, the graph below shows a parabola, indicating that the equation is of the form y=ax^{2}+bx+c. By finding the x- and y-intercepts, one can determine the constants and complete the equation: y=2x^{2}+4x-1.

In conclusion, completing an equation for the function graphed above requires one to first identify the type of function, and then use the graph to determine the constants of the equation. With a bit of practice, one can become proficient in completing equations for functions graphed above.

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