Understanding Linear Functions: Definition, Graphing, and Applications

Linear Function

A linear function is a mathematical function that describes a straight line

A linear function is a mathematical function that describes a straight line. It has the form of y = mx + b, where m represents the slope of the line and b represents the y-intercept.

The slope (m) determines how steep the line is. A positive slope means the line goes upward from left to right, while a negative slope means the line goes downward. A slope of zero means the line is horizontal.

The y-intercept (b) represents the point where the line crosses the y-axis. It is the value of y when x is equal to zero.

To graph a linear function, you can use the slope and the y-intercept. Start by plotting the y-intercept on the y-axis. Then, using the slope, determine another point on the line. For example, if the slope is 2, for every 1 unit increase in x, the y-value will increase by 2. So you can plot another point by moving 1 unit to the right and 2 units up (or down depending on the slope). Connect these two points with a straight line to graph the linear function.

To find the x-intercept (the point where the line crosses the x-axis), you need to set y equal to zero and solve the equation. For example, if you have the linear function y = 3x – 2, you can set y equal to zero and solve for x: 0 = 3x – 2. Rearrange the equation to isolate x: 3x = 2, and then divide both sides by 3: x = 2/3. Therefore, the x-intercept is (2/3, 0).

To find the y-intercept, you can set x equal to zero: y = 3(0) – 2. Simplify the equation to find that the y-intercept is (0, -2).

Linear functions are widely used to model real-world situations, such as determining the total cost of a purchase based on the number of items bought, or estimating future values based on known data points.

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