A linear function is a mathematical function whose graph forms a straight line. The standard form of a linear function is y equals m x plus b, where m represents the slope of the line and b represents the y-intercept, which is where the line crosses the y-axis.
The slope m is a crucial parameter that determines both the steepness and direction of a linear function. A positive slope means the line rises from left to right, while a negative slope means it falls. A zero slope creates a horizontal line, and an undefined slope results in a vertical line. The slope can be calculated using the rise over run formula.
The y-intercept b represents the point where the linear function crosses the y-axis. This occurs when x equals zero. A positive b means the line crosses above the origin, a negative b means it crosses below, and when b equals zero, the line passes through the origin. To find the y-intercept, simply substitute x equals zero into the equation.
To graph a linear function, follow these steps: First, identify the slope m and y-intercept b from the equation. Second, plot the y-intercept point at coordinates zero comma b. Third, use the slope to find the next point by moving according to rise over run. Finally, draw a straight line through these points. Let's see this with the example y equals negative one half x plus three.
To summarize what we have learned about linear functions: They create straight line graphs and follow the standard form y equals m x plus b. The slope m controls both the steepness and direction of the line, while the y-intercept b determines where the line crosses the y-axis. Linear functions are fundamental building blocks in mathematics and appear in many real-world applications.