Transformations of Functions: Stretching, Shrinking, and Vertical Translation

Transformations of Functions

Stretching and Shrinking of Functions

Transformations such as stretching and shrinking change the shape of a function's graph by multiplying the function by a constant. These transformations are important in understanding how graphs behave under scaling.

• Vertical Stretch: If $a > 1$, the graph of $y = f(x)$ is stretched vertically by a factor of $a$.

• Vertical Shrink: If $0 < a < 1$, the graph of $y = f(x)$ is shrunk vertically by a factor of $a$.

• Reflection and Stretch/Shrink: If $a < 0$, the graph is reflected across the x-axis and also stretched or shrunk depending on the absolute value of $a$.

Example: Consider the functions $f(x) = \sqrt{x}$, $g(x) = 2\sqrt{x}$, and $h(x) = \frac{1}{2}\sqrt{x}$.

• $f(x) = \sqrt{x}$: The basic square root function.

• $g(x) = 2\sqrt{x}$: The graph is stretched vertically by a factor of 2.

• $h(x) = \frac{1}{2}\sqrt{x}$: The graph is shrunk vertically by a factor of $\frac{1}{2}$.

Graphical Comparison: The graph shows how each transformation affects the shape and position of the square root function. For example, $g(x)$ rises more steeply than $f(x)$, while $h(x)$ rises more slowly.

Vertical Translation

A vertical translation shifts the graph of a function up or down without changing its shape. This is achieved by adding or subtracting a constant to the function.

• Definition: If $k > 0$, then the graph of $y = f(x) + k$ is a translation of $k$ units upward. If $k < 0$, then the graph of $y = f(x) + k$ is a translation of $k$ units downward.

Example: Graph $f(x) = |x|$, $g(x) = |x| + 2$, and $h(x) = |x| - 2$.

• $f(x) = |x|$: The basic absolute value function.

• $g(x) = |x| + 2$: The graph is shifted 2 units upward.

• $h(x) = |x| - 2$: The graph is shifted 2 units downward.

Summary Table: Effects of Vertical Stretch/Shrink and Translation

Transformation	Equation	Effect on Graph
Vertical Stretch	$y = a f(x)$, $a > 1$	Stretches graph vertically by factor $a$
Vertical Shrink	$y = a f(x)$, $0 < a < 1$	Shrinks graph vertically by factor $a$
Vertical Translation Up	$y = f(x) + k$, $k > 0$	Shifts graph up by $k$ units
Vertical Translation Down	$y = f(x) + k$, $k < 0$	Shifts graph down by $k$ units

Additional info: These transformations are foundational in College Algebra and are used to analyze and graph a wide variety of functions, including polynomial, radical, and absolute value functions.