Final answer:
Linear functions, represented by y = mx + b, model relationships with a constant rate of change. Slope (m) represents the rate, while y-intercept (b) indicates where the line crosses the y-axis. Evaluating, writing rules for, and graphically analyzing linear functions equip students with tools to interpret and manipulate these relationships.
Step-by-step explanation:
Linear functions are foundational in mathematics and serve as a model for understanding relationships between variables that change at a constant rate. A linear function is typically written in the form y = mx + b, where m is the slope of the line and b is the y-intercept. To evaluate a linear function, you simply replace the variable 'x' with a specific value to find the corresponding 'y' value. Writing linear function rules involves identifying the slope and y-intercept from a table or graph and constructing the function's equation. A graphical analysis of linear functions will show a straight line on the Cartesian plane, and the slope of the line indicates how steep the line is, while the intercept shows where the line crosses the y-axis.
When manipulating a linear function, changing the slope will tilt the line steeper or flatter, while changing the y-intercept will shift the line up or down the graph. Computing and interpreting a growth rate involves looking at the slope as a percentage change; for example, a slope of 0.2 can be interpreted as a 20% increase for every unit increase in the x-value. By analyzing graphical representations and working with function notation, students can better understand how linear functions model real-world situations.