Horizontal multiplication

Functions and graphs: Transformations of graphs and functions

Horizontal multiplication

For any function $f$ and any number $b\neq 0$ , the graph of the function $g$ defined by $g(x)= f(b\cdot x)$ is obtained from the graph of $f$ by multiplying it horizontally by $1/b$ : the horizontal coordinate is multiplied by $1/b$ .

If $b>0$ , then it is about stretching or shrinking the graph relative to the vertical axis.
If $b<0$ , then all points will be on the other side of the vertical axis.
A special case is $b=-1$ : then the graph of $f$ mirrored in the vertical axis.

If $\bigl(x,f(x)\bigr)$ a point on the graph of $f$ , then $\bigl(x/b,g(x/b)\bigr)=\bigl(x/b,f(x)\bigr)$ is a point on the graph of $g$ : for the same vertical coordinate, the horizontal coordinate is divided by $b$ .

The figure below shows the graphs of $x\mapsto x(x+1)(x-2)$ , $x \mapsto 2x(2x+1)(2x-2)$ and $x \mapsto -x(-x+1)(-x-2)$ .

You get the red graph of $x \mapsto 2x(2x+1)(2x-2)$ by multiplying the blue graph of $x\mapsto x(x+1)(x-2)$ horizontally by a factor of $\frac{1}{2}$

You get the green graph of $x \mapsto -x(-x+1)(-x-2)$ by mirroring the blue graph of $x\mapsto x(x+1)(x-2)$ in the vertical axis.

horizontale vermenigvuldiging