When you apply a *vertical** transformation* to a parent graph, you are stretching or shrinking the graph along the *y**-*axis, which changes its height. A number (or *coefficient*) multiplying in front of a function causes the vertical transformation. The coefficient always affects the height of each and every point in the graph of the function. We call the vertical transformation a *stretch* if the coefficient is greater than 1 and a *shrink* if the coefficient is between 0 and 1.

For example, the graph of *f*(*x*) = 2*x*^{2} takes the graph of *f*(*x*) = *x*^{2} and stretches it by a vertical factor of two. That means that each time you plot a point vertically on the graph, the value gets multiplied by two (making the graph twice as tall at each point). For example, in *f(x)=x** ^{2}*, 1 gets mapped to 2×1

^{2}=2, 2 gets mapped to 2×2

^{2}=8, 3 gets mapped to 2×3

^{2}=18, etc.

The vertical transformation of *f*(*x*) = 2*x*^{2} and

are shown in this figure.

The transformation** **rules apply to

*any*function, so the vertical transformation of

is shown here.

The 4 is a vertical stretch; it makes the graph four times as tall at every point. For example, 1 gets mapped to 4×sqrt(1)=4, 4 gets mapped to 4×sqrt(4)=8, 9 gets mapped to 4×sqrt(9)=12, etc. (notice that this example uses numbers that you can easily take the square root of to make graphing a simple task); and so on.