Stretches

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

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

The stretch is centered on the x-axis, meaning any point where $y=0$ (the x-intercepts) will remain stationary.

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

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

In this transformation, the x-coordinates of all points are multiplied by $\frac{1}{b}$, while the y-coordinates remain unchanged.

The stretch is centered on the y-axis, meaning any point where $x=0$ (the y-intercept) will remain stationary.

If $|b| > 1$, the graph is horizontally compressed toward the y-axis; if $0 < |b| < 1$, the graph is horizontally stretched away from the y-axis.

Identify Invariant Points: Always check where the graph crosses the axis of the stretch. For $y = a \cdot f(x)$, x-intercepts stay the same. For $y = f(b \cdot x)$, the y-intercept stays the same.

Reciprocal Rule: For horizontal transformations, remember the 'inside-out' rule: operations inside the function brackets affect $x$ and often act inversely (e.g., multiplying by 2 results in a scale factor of $1/2$).

Asymptote Behavior: If a graph has asymptotes, they are stretched just like points. A horizontal asymptote $y=k$ will move to $y=a \cdot k$ under a vertical stretch, but a vertical asymptote $x=h$ will remain unchanged.

Sanity Check: If the scale factor is 2, the graph should look 'taller' (vertical) or 'wider' (horizontal). If it looks 'thinner' or 'shorter', you may have applied the factor to the wrong axis.

Confusing the Horizontal Factor: The most common error is using $b$ as the horizontal scale factor instead of $\frac{1}{b}$. For $y = f(3x)$, the graph is 3 times narrower, not 3 times wider.

Stretching the Origin: Students often forget that $(0,0)$ is invariant for both types of stretches. If a graph passes through the origin, that point never moves during a simple stretch.

Order of Operations: When combining stretches with translations, the order matters significantly. Usually, stretches are applied before translations unless brackets indicate otherwise.