Stretches of Functions

• Definition of a Stretch: A stretch is a transformation that moves all points on a graph towards or away from a specific axis (either the x-axis or the y-axis) by a constant factor. Unlike translations, stretches change the size and shape of the graph, making it appear compressed or elongated.

• Scale Factor: The scale factor, often denoted by $a$, determines the magnitude of the stretch. If $a > 1$, the graph is stretched away from the axis; if $0 < a < 1$, the graph is compressed towards the axis. A negative scale factor would imply a reflection combined with a stretch, which is typically treated as a separate reflection transformation.

• Axis of Stretch: Every stretch occurs relative to an axis, known as the axis of stretch. For vertical stretches, the x-axis serves as the axis of stretch, meaning points on the x-axis remain fixed. For horizontal stretches, the y-axis is the axis of stretch, and points on the y-axis remain fixed.

• Equation Form: A vertical stretch of a function $y = f(x)$ is represented by the equation $y = af(x)$. Here, the output values of the function are multiplied by the scale factor $a$.

• Effect on Coordinates: For any point $(x, y)$ on the original graph $y = f(x)$, the corresponding point on the stretched graph $y = af(x)$ will be $(x, ay)$. The x-coordinates remain unchanged, while the y-coordinates are scaled by $a$.

• Axis of Stretch and Fixed Points: The x-axis ($y=0$) acts as the axis of stretch for vertical transformations. Any point that lies on the x-axis (i.e., where $y=0$) will remain fixed, as $a \cdot 0 = 0$. All other points move parallel to the y-axis, either away from the x-axis (if $a > 1$) or towards it (if $0 < a < 1$).

• Example: If the graph of $y = x^2$ is vertically stretched by a factor of 2, the new equation is $y = 2x^2$. A point like $(1, 1)$ on $y=x^2$ becomes $(1, 2)$ on $y=2x^2$.

• Equation Form: A horizontal stretch of a function $y = f(x)$ is represented by the equation $y = f(ax)$. In this case, the input values to the function are scaled before the function is evaluated.

• Effect on Coordinates: For any point $(x, y)$ on the original graph $y = f(x)$, the corresponding point on the stretched graph $y = f(ax)$ will be $(\frac{x}{a}, y)$. The y-coordinates remain unchanged, while the x-coordinates are scaled by a factor of $\frac{1}{a}$. This inverse relationship means that if $a > 1$, the graph is compressed horizontally, and if $0 < a < 1$, it is stretched horizontally.

• Axis of Stretch and Fixed Points: The y-axis ($x=0$) serves as the axis of stretch for horizontal transformations. Any point that lies on the y-axis (i.e., where $x=0$) will remain fixed, as $f(a \cdot 0) = f(0)$. All other points move parallel to the x-axis, either away from the y-axis (if $0 < a < 1$) or towards it (if $a > 1$).

• Example: If the graph of $y = \sin(x)$ is horizontally compressed by a factor of 2 (meaning $a=2$), the new equation is $y = \sin(2x)$. A point like $(\pi/2, 1)$ on $y=\sin(x)$ becomes $(\pi/4, 1)$ on $y=\sin(2x)$.

• Input vs. Output Transformation: The key to understanding stretches lies in whether the transformation affects the input ($x$) or the output ($y$) of the function. When the function's output $f(x)$ is multiplied by a factor $a$ (as in $y=af(x)$), it directly scales the y-coordinates, leading to a vertical stretch.

• Inverse Effect on Input: When the input $x$ is multiplied by a factor $a$ inside the function (as in $y=f(ax)$), the effect on the graph's x-coordinates is inverse. To achieve the same output $y$, the new input $ax'$ must equal the original input $x$, meaning $x' = x/a$. This results in the x-coordinates being scaled by $\frac{1}{a}$, causing a horizontal stretch or compression.

• Preservation of Shape: While stretches change the size and overall dimensions of the graph, they preserve its fundamental shape characteristics. For instance, a parabola remains a parabola, and a sine wave remains a sine wave, though its amplitude or period might change.

• Transforming Asymptotes: Asymptotes are lines that the graph approaches but never touches. When a function is stretched, its asymptotes are also transformed according to the same rules as the points on the graph. For example, a vertical asymptote $x=c$ will become $x=c/a$ under a horizontal stretch $y=f(ax)$, and a horizontal asymptote $y=d$ will become $y=ad$ under a vertical stretch $y=af(x)$.

• Exceptions for Asymptotes: If an asymptote is parallel to the direction of the stretch, or if it coincides with the axis of stretch, it will not be affected. For instance, a vertical asymptote $x=0$ (the y-axis) is unaffected by a horizontal stretch $y=f(ax)$ because the y-axis is the axis of stretch. Similarly, a horizontal asymptote $y=0$ (the x-axis) is unaffected by a vertical stretch $y=af(x)$.

• Transforming Intercepts: Intercepts are points where the graph crosses the coordinate axes. An x-intercept $(x_0, 0)$ will remain an x-intercept $(x_0, 0)$ under a vertical stretch $y=af(x)$, but will become $(x_0/a, 0)$ under a horizontal stretch $y=f(ax)$. Conversely, a y-intercept $(0, y_0)$ will become $(0, ay_0)$ under a vertical stretch $y=af(x)$, but will remain $(0, y_0)$ under a horizontal stretch $y=f(ax)$.

Understanding the differences between vertical and horizontal stretches is crucial for correctly applying transformations.

• Identify the Transformation Type: Always begin by clearly identifying whether the given transformation is a vertical stretch ($y = af(x)$) or a horizontal stretch ($y = f(ax)$). This dictates which coordinate (x or y) is affected and by what factor.

• Determine the Scale Factor: For $y = af(x)$, the scale factor is simply $a$. For $y = f(ax)$, remember the scale factor for the x-coordinates is $\frac{1}{a}$. A common mistake is to use $a$ directly for horizontal stretches, leading to incorrect transformations.

• Transform Key Points: When sketching a stretched graph, always transform any marked points from the original graph. Apply the coordinate rules: $(x, y) \to (x, ay)$ for vertical stretches, and $(x, y) \to (x/a, y)$ for horizontal stretches.

• Handle Intercepts and Asymptotes: Pay special attention to how intercepts and asymptotes are affected. X-intercepts are invariant under vertical stretches, and y-intercepts are invariant under horizontal stretches. Remember the rules for transforming asymptotes, noting when they remain unaffected.

• Visualize the Effect: Before drawing, mentally visualize the effect of the stretch. Will the graph become taller or shorter? Wider or narrower? This helps in performing a quick sanity check on your transformed points and overall sketch.