Linear Functions: Understanding the Basics, Equations, and Graphing

Linear Function

A linear function is a type of mathematical function that represents a straight line on a graph

A linear function is a type of mathematical function that represents a straight line on a graph. It is characterized by having a constant rate of change, or slope, between any two points on the line. In other words, for every unit increase in the independent variable, there is a constant change in the dependent variable.

The equation of a linear function is typically written in the form:

y = mx + b

where:
– y represents the dependent variable (usually the y-coordinate on a graph)
– x represents the independent variable (usually the x-coordinate on a graph)
– m represents the slope of the line, which determines how steep or slanted the line is
– b represents the y-intercept, which is the point where the line crosses the y-axis

To graph a linear function, we need at least two points. A convenient way to find these points is to set x to specific values and calculate the corresponding y values using the given equation.

For example, let’s say we have the equation y = 2x + 3. We can choose two different x values, such as x = 0 and x = 2, and find the corresponding y values:

When x = 0, y = 2(0) + 3 = 3. So one point on the graph is (0, 3).
When x = 2, y = 2(2) + 3 = 7. So another point on the graph is (2, 7).

We can then plot these points on a graph and draw a straight line passing through them. The slope of the line is m = 2, which means for every one unit increase in x, the corresponding y value increases by 2. The y-intercept is b = 3, so the line crosses the y-axis at (0, 3).

In addition to graphing linear functions, we can also perform other operations such as finding the slope between two points on the line, determining the x-intercept and y-intercept, and solving equations involving linear functions.

Overall, linear functions are fundamental in mathematics and have numerous applications in various fields such as physics, engineering, economics, and more.

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