A linear function is a function whose graph is a straight line. It can be expressed in the form:
f(x)=mx+bf(x)=mx+bf(x)=mx+b
where:
- mmm is the slope of the line, representing the rate of change of the function,
- bbb is the y-intercept, the point where the line crosses the y-axis,
- xxx is the independent variable,
- f(x)f(x)f(x) or yyy is the dependent variable.
The slope mmm indicates how steep the line is: if m>0m>0m>0, the line slopes upward; if m<0m<0m<0, it slopes downward; and if m=0m=0m=0, the function is constant and the graph is a horizontal line
. In more variables, a linear function can be written as:
f(x1,x2,…,xk)=b+a1x1+a2x2+⋯+akxkf(x_1,x_2,\ldots,x_k)=b+a_1x_1+a_2x_2+\cdots +a_kx_kf(x1,x2,…,xk)=b+a1x1+a2x2+⋯+akxk
which represents a hyperplane in higher dimensions
. In summary, a linear function is an algebraic function of degree zero or one that produces a straight line when graphed. It models relationships where the output changes at a constant rate with respect to the input
