Linear functions

Definition

Linear functions are mathematical expressions that create a straight line when graphed on a coordinate plane. They are defined by the equation $$f(x) = mx + b$$, where $$m$$ represents the slope and $$b$$ is the y-intercept. The characteristics of linear functions, such as their constant rate of change and their representation as straight lines, allow them to be easily composed with other functions to produce new outputs.

5 Must Know Facts For Your Next Test

1. The graph of a linear function is always a straight line, which indicates a constant rate of change between variables.
2. Linear functions can be represented in different forms, such as slope-intercept form ($$y = mx + b$$), point-slope form, and standard form ($$Ax + By = C$$).
3. The slope of a linear function determines its direction; a positive slope indicates an upward trend, while a negative slope shows a downward trend.
4. When composing linear functions, the result is always another linear function, making it easy to analyze complex relationships.
5. The domain and range of linear functions are typically all real numbers unless restricted by context.

Review Questions

• How do you determine the slope and y-intercept of a linear function from its equation?
  • To find the slope and y-intercept of a linear function in slope-intercept form ($$y = mx + b$$), you simply identify the coefficient $$m$$ as the slope and the constant $$b$$ as the y-intercept. The slope indicates how steep the line is and whether it rises or falls as you move along the x-axis. The y-intercept is where the line crosses the y-axis, giving you a starting point for graphing.
• What happens to the composition of two linear functions in terms of their graphical representation?
  • When two linear functions are composed, the result is another linear function. This means that if you take one linear function and apply it to another, you'll still get a straight line when graphed. The characteristics of linearity remain intact, allowing you to analyze how changes in one variable affect another in a consistent manner. The composition will have its own slope and y-intercept based on the original functions.
• Evaluate how understanding linear functions aids in solving real-world problems through function composition.
  • Understanding linear functions allows for effective modeling of real-world situations where relationships between variables are constant. When composing linear functions, you can create more complex models that reflect how different factors interact. For instance, in economics, you might model supply and demand as two separate linear functions and then compose them to analyze market equilibrium. This ability to combine functions helps clarify relationships and can lead to better decision-making based on quantitative data.

Related terms

slope: The slope is the ratio of the vertical change to the horizontal change between two points on a line, indicating its steepness.

y-intercept: The y-intercept is the point at which a line crosses the y-axis, representing the output value when the input value is zero.

function composition: Function composition is the process of applying one function to the results of another function, allowing for the creation of more complex functions.