Linear Functions
Overview
Linear functions are fundamental in mathematics, characterized by their representation as straight lines on a graph. This document explains how to identify a linear function, determine its slope, and find the y-intercept.
Learning Objectives
- Recognize linear function descriptions.
- Plot linear function graphs.
- Determine the slope and y-intercept of a linear function graph.
Basic Form
A linear function is generally expressed as:
- f(x) = ax + b
- a: gradient (slope) of the line
- b: y-intercept (vertical intercept)
Variable Analysis
- Fixing b: Varying a affects the line's slope:
- If a > 0, the line ascends with increasing x.
- If a < 0, the line descends.
- Fixing a: Varying b shifts the line up or down without changing its slope.
Special Cases
- a = 0: Results in horizontal lines crossing the y-axis at b.
- b = 0: Lines pass through the origin with slope a.
Non-standard Form
Linear functions can also appear in forms that require rearrangement to the standard form:
- Example: 4x - 3y = 2 rearranges to y = (4/3)x - 2/3
- Key is making y the subject to identify a and b.
Key Points
- Functions in the form f(x) = ax + b are linear.
- a: Slope, dictates the angle of inclination.
- b: Y-intercept, indicates where the line crosses the y-axis.
Exercises
- Define a linear function.
- Plot linear functions using given equations on a graph.
- Rearrange equations to identify gradients and intercepts.
- Create functions with parallel lines (same slope).
- Create functions with the same y-intercept.
Examples
- Equations and corresponding gradients and intercepts are provided, reinforcing the concepts of slope and y-intercept.
- Understanding through practice is emphasized, ensuring comprehension of linear function plotting and modification from non-standard forms.
Conclusion
Linear functions, characterized by the formula f(x) = ax + b, are straightforward and vital for understanding linear relationships in mathematics. Mastery requires practice in plotting and rearrangement.