key term - Vertical Stretch

Definition

Vertical stretch is a transformation of a function that involves scaling the function vertically, either by expanding or compressing the graph along the y-axis. This transformation affects the amplitude or range of the function, altering its visual appearance without changing its overall shape or behavior.

5 Must Know Facts For Your Next Test

1. Vertical stretch is represented mathematically by multiplying the function $f(x)$ by a constant $a$, where $a > 0$, to create a new function $g(x) = a \cdot f(x)$.
2. When $a > 1$, the function is stretched vertically, and when $0 < a < 1$, the function is compressed vertically.
3. Vertical stretch affects the amplitude or range of the function, but it does not change the function's period or frequency.
4. Vertical stretch is a common transformation used in the analysis of various function types, including absolute value functions, power functions, polynomial functions, logarithmic functions, and trigonometric functions.
5. Understanding vertical stretch is crucial for accurately sketching and interpreting the graphs of these functions, as well as for solving related problems involving transformations.

Review Questions

• Explain how vertical stretch affects the graph of an absolute value function.
  • For an absolute value function $f(x) = |x|$, a vertical stretch by a factor of $a$ results in the function $g(x) = a \cdot |x|$. This transformation changes the amplitude or range of the function, making the graph wider or narrower along the y-axis, while maintaining the overall V-shape of the original absolute value function.
• Describe the effect of vertical stretch on the graph of a power function.
  • For a power function $f(x) = x^n$, where $n$ is a positive integer, a vertical stretch by a factor of $a$ results in the function $g(x) = a \cdot x^n$. This transformation changes the scale of the function along the y-axis, making the graph taller or shorter, without altering the basic shape of the power function, which is determined by the exponent $n$.
• Analyze the impact of vertical stretch on the graph of a logarithmic function.
  • For a logarithmic function $f(x) = \log_b(x)$, where $b > 0$ and $b \neq 1$, a vertical stretch by a factor of $a$ results in the function $g(x) = a \cdot \log_b(x)$. This transformation changes the amplitude or range of the logarithmic function, making the graph taller or shorter, while preserving the overall concave-down shape and the horizontal asymptote at $y = 0$.

Related terms

Transformation of Functions: Transformations of functions involve modifying the original function to create a new function with different characteristics, such as changes in the domain, range, or graph.

Amplitude:

The amplitude of a function is the distance between the midline and the maximum or minimum values of the function, representing the function's vertical scale.

Vertical Shift:

Vertical shift is a transformation that moves the graph of a function up or down along the y-axis, without changing its shape or size.