Linear functions

Summary

Definition

• Given: a real-valued function of a real variable, \(f: R→R\)
• We say that \(f\) is a linear function if there are constants \(a,b\) such that

\[f(x)=ax+b\]

• for all \(x∈R\)
• \(b\) is called the intercept of the function and \(a\) is called the slope of the function.
• The domain (and the codomain) may be subsets of \(R\) .

Some results

• The graph of a linear function is a straight line .
• If \(a > 0\) then \(f\) is strictly increasing and the line slants upwards to the right. The larger the value of \(a\) , the steeper is the line.
• If \(a=0\) then \(f(x)=b\) and the line is horizontal.
• If \(a < 0\) then \(f\) is strictly decreasing and the line slants downwards to the right. The larger the absolute value of \(a\) , the steeper is the line.

Slope of a linear function

• \(f\) is a linear function, \(x_1,x_2\) are two points in the domain, \(x_1≠x_2\) and \(y_1=f\left( x_1 \right), y_2=f\left( x_2 \right)\) .
• Then \((x_1,y_1)\) and \((x_2,y_2)\) are two distinct points on the graph of \(f\) .
• We have

\[a= \frac{y_2-y_1}{x_2-x_1}= \frac{Δy}{Δx}\]

• where \(Δy=y_2-y_1\) and \(Δx=x_2-x_1\)
• For \(Δx=1\) , \(a=Δy\) . \(y\) increases by \(a\) units when \(x\) increases by 1 unit.

The equation of the straight line

• The equation of the straight line passing through \((x_1,y_1)\) with slope \(a\) is given by

\[y-y_1=a(x-x_1)\]