The Linear Function and Its Graph
Definition of \(f(x) = ax + b\), its graph as a line, slope and intercept, monotonicity, and the sign of the linear function.
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Learning Objectives
- State the definition of a linear function \(f \colon \mathbb{R} \to \mathbb{R},\; f(x) = ax + b,\; a \neq 0\).
- Distinguish the affine case from direct proportionality \((b = 0)\), recalling linear growth from arithmetic progressions.
- Draw the graph of a linear function and identify slope and intercept.
- Determine monotonicity from the sign of \(a\).
- Build the sign table of a linear function, with its zero at \(x = -\tfrac{b}{a}\).
About This Course Unit
We define the linear function and study its graph: a line determined by two points, with slope \(a\) and intercept \(b\). From there we read off its monotonicity and its sign.