Vertical multiplication

Functions and graphs: Transformations of graphs and functions

Vertical multiplication

For any function $f$ and any number $a\neq 0$ , the graph of the function $g$ defined by $g(x)=a\cdot f(x)$ is obtained from the graph of $f$ by multiplying it vertically by $a$ : the vertical coordinate is multiplied by $a$ .

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

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

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

The red graph of $x \mapsto 2x(x+1)(x-2)$ is obtained by stretching the blue graph of $x\mapsto x(x+1)(x-2)$ vertically by a factor of $2$ ( $a=2$ ).

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 horizontal ( $a=-1$ ).

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