Stretches

A vertical stretch is represented by the transformation $y = af(x)$, where $a$ is the scale factor.

In this transformation, the x-coordinates of all points remain unchanged, while the y-coordinates are multiplied by $a$.

If $a > 1$, the graph is pulled away from the x-axis; if $0 < a < 1$, the graph is compressed toward the x-axis.

Points located on the x-axis (where $y=0$) are invariant points because $a \cdot 0 = 0$.

A horizontal stretch is represented by the transformation $y = f(ax)$, where the scale factor is $\frac{1}{a}$.

This transformation is often counter-intuitive: multiplying the input $x$ by $a$ results in a stretch by the reciprocal $\frac{1}{a}$.

The y-coordinates remain unchanged, while the x-coordinates are multiplied by $\frac{1}{a}$.

If $a > 1$, the graph is compressed toward the y-axis (scale factor $< 1$); if $0 < a < 1$, the graph is stretched away from the y-axis (scale factor $> 1$).

Vertical stretches affect the 'outside' of the function (the output), while horizontal stretches affect the 'inside' (the input).

Always remember that horizontal transformations typically act in the opposite way to what the operation suggests (multiplication by $a$ results in division by $a$ for the coordinates).

Asymptotes are treated like any other line on the graph and are subject to the same coordinate transformations.

A vertical asymptote ($x = k$) is affected by horizontal stretches, moving to $x = \frac{k}{a}$, but is unaffected by vertical stretches.

A horizontal asymptote ($y = c$) is affected by vertical stretches, moving to $y = ac$, but is unaffected by horizontal stretches.

If an asymptote lies exactly on the invariant axis (e.g., the x-axis for a vertical stretch), its position remains unchanged.