Linear Function

Definition

A linear function is a mathematical expression that describes a relationship between two variables where the graph of the equation forms a straight line. This relationship can be expressed in the form of an equation, typically written as $$y = mx + b$$, where $$m$$ is the slope and $$b$$ is the y-intercept. The linearity implies that there is a constant rate of change between the variables, which means that for every unit increase in one variable, there is a proportional change in the other variable.

5 Must Know Facts For Your Next Test

1. Linear functions can be identified by their equations being in first-degree polynomial form, meaning that each variable is raised to the power of one.
2. The graph of a linear function will always produce a straight line, regardless of its slope or y-intercept.
3. The slope of a linear function indicates whether the function is increasing or decreasing; a positive slope means it's increasing, while a negative slope means it's decreasing.
4. In real-world applications, linear functions are often used to model relationships where changes are consistent, such as calculating costs or predicting trends.
5. A horizontal line represents a linear function with a slope of zero, indicating no change in y regardless of changes in x.

Review Questions

• How can you determine if a given equation represents a linear function? What characteristics should you look for?
  • To determine if an equation represents a linear function, you should check if it can be rearranged into the standard form $$y = mx + b$$. Key characteristics include that it should only contain first-degree terms (no powers higher than one) and no products or functions involving the variables. Additionally, if graphed, it should yield a straight line. If any of these conditions are violated, then it does not represent a linear function.
• Describe how changes in slope and y-intercept affect the graph of a linear function.
  • Changes in slope directly affect how steeply the line rises or falls; increasing the slope makes the line steeper, while decreasing it makes it flatter. The y-intercept determines where the line crosses the y-axis; shifting this value up or down will move the entire line vertically. Together, these two parameters define all aspects of the graph’s appearance and position in relation to both axes.
• Evaluate how understanding linear functions can enhance your ability to model real-world scenarios effectively.
  • Understanding linear functions equips you with valuable tools for modeling situations that exhibit consistent relationships between variables, such as budgeting or speed over time. By knowing how to manipulate and interpret linear equations and graphs, you can make accurate predictions and decisions based on data trends. This insight into slopes and intercepts can help you analyze rates of change, optimize processes, and convey information clearly through visual representations.

Related terms

Slope: The slope is a measure of the steepness of a line, calculated as the change in the y-coordinate divided by the change in the x-coordinate between two points on a line.

Y-Intercept: The y-intercept is the point where the graph of a function intersects the y-axis, which represents the value of the function when the input (x) is zero.

Function Notation: Function notation is a way to represent functions using symbols, often denoted as $$f(x)$$, where $$f$$ indicates the function and $$x$$ represents the input variable.