Understanding Linear Functions: Definition, Graphing, and Applications

linear function

A linear function is a type of mathematical function that can be represented by a straight line on a graph

A linear function is a type of mathematical function that can be represented by a straight line on a graph. It has the form f(x) = mx + b, where m is the slope and b is the y-intercept.

The slope, represented by m, determines how steep the line is. If m is positive, the line will rise as x increases. If m is negative, the line will fall as x increases. The absolute value of m represents the steepness of the line.

The y-intercept, represented by b, is the point at which the line crosses the y-axis. It is the value of f(x) when x = 0.

To graph a linear function, you can plot the y-intercept (0, b) and then use the slope to find other points. If the slope is a fraction, you can use rise over run to determine the change in y and x values. For example, if the slope is 2/3, for every increase of 3 in x, the corresponding y will increase by 2.

You can also use the slope-intercept form, y = mx + b, to identify the slope and y-intercept of a linear function from its equation. This form makes it easier to find other points to graph.

Linear functions are often used to model relationships between two variables that are directly proportional to each other. For example, the temperature in Celsius can be converted to Fahrenheit using the linear function F = (9/5)C + 32, where F represents Fahrenheit and C represents Celsius.

In summary, a linear function is a mathematical function that represents a straight line on a graph. It has the form f(x) = mx + b, where m is the slope and b is the y-intercept. The slope determines the line’s steepness, and the y-intercept is where the line crosses the y-axis. Linear functions are commonly used to model proportional relationships between variables.

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