A linear function is a mathematical function that can be written in the form f of x equals m x plus b, where m represents the slope and b represents the y-intercept. Linear functions are called linear because they create straight lines when graphed on a coordinate plane. The slope m determines how steep the line is and in which direction it goes, while the y-intercept b tells us where the line crosses the y-axis. For example, the function f of x equals 2x plus 1 has a slope of 2 and a y-intercept of 1, creating a straight line that rises 2 units for every 1 unit it moves to the right.
The slope parameter m in a linear function determines both the steepness and direction of the line. Slope is calculated as rise over run, or the change in y divided by the change in x. A positive slope means the line goes upward from left to right, while a negative slope means it goes downward. The larger the absolute value of the slope, the steeper the line becomes. For example, a line with slope 2 rises steeply upward, while a line with slope negative 3 falls steeply downward. A slope of 0.5 creates a gentle upward line, and a slope of 0 creates a horizontal line. Understanding slope helps us predict how quickly one variable changes with respect to another.
The y-intercept parameter b determines where the line crosses the y-axis. This occurs at the point zero comma b, where x equals zero. The y-intercept represents the starting value or initial condition in many real-world applications. When we change the y-intercept while keeping the slope constant, we create parallel lines that are shifted vertically. For example, the functions f of x equals 2x plus 3, f of x equals 2x, and f of x equals 2x minus 2 all have the same slope of 2, making them parallel. However, they cross the y-axis at different points: 3, 0, and negative 2 respectively. This vertical shifting effect allows us to model situations where the rate of change remains constant but the starting point varies.
Graphing linear functions follows a systematic four-step process. First, identify the slope m and y-intercept b from the function. Second, plot the y-intercept point at coordinates zero comma b on the y-axis. Third, use the slope to find a second point by moving from the y-intercept according to the rise over run pattern. Finally, draw a straight line through both points. Let's demonstrate with the example f of x equals negative 3x plus 4. The slope is negative 3 and the y-intercept is 4. We start by plotting the point zero comma 4. Since the slope is negative 3, we move down 3 units and right 1 unit from our first point, reaching the coordinates 1 comma 1. Drawing a line through these two points gives us the complete graph of our linear function.
Linear functions appear frequently in real-world applications. One common example is cost calculation, where there's a fixed fee plus a variable rate. Consider a phone plan that costs 30 dollars per month plus 10 cents per minute of usage. This can be represented as C of m equals 0.10 times m plus 30, where m is minutes and C is total cost. The y-intercept of 30 represents the fixed monthly fee, while the slope of 0.10 represents the cost per minute. Other applications include temperature conversion, where Fahrenheit equals nine-fifths times Celsius plus 32, and distance-time relationships for constant speed travel, such as distance equals 60 times time for a car traveling at 60 miles per hour. These linear relationships help us model and predict outcomes in many practical situations.