Vertical compression is a transformation of a function that scales the function vertically, either stretching or shrinking it along the y-axis. This transformation affects the amplitude or range of the function, without changing its horizontal properties or period.
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Vertical compression of a function $f(x)$ is achieved by multiplying the function by a scale factor $a$, where $0 < a < 1$, resulting in the transformed function $f(x) = a \cdot f(x)$.
Vertical compression reduces the amplitude or range of a function, making it appear more 'squished' along the y-axis.
Vertical compression affects the graphs of various function types, including absolute value functions, power functions, polynomial functions, exponential functions, logarithmic functions, and trigonometric functions.
Vertical compression can be used to model real-world phenomena, such as the relationship between input and output in certain physical or economic systems.
Vertical compression, along with other transformations like translation and reflection, is a key concept in the study of function transformations and their applications.
Review Questions
Explain how vertical compression affects the graph of an absolute value function.
When an absolute value function $f(x) = |x|$ undergoes vertical compression, the resulting function $f(x) = a \cdot |x|$, where $0 < a < 1$, will have a smaller amplitude or range compared to the original function. The graph of the compressed absolute value function will appear 'squished' vertically, with the peak or vertex remaining at the same horizontal position, but the function's maximum and minimum values being scaled down by the factor $a$.
Describe the effect of vertical compression on the graph of a power function.
For a power function $f(x) = x^n$, where $n$ is a positive integer, vertical compression by a factor $a$, where $0 < a < 1$, results in the transformed function $f(x) = a \cdot x^n$. This transformation scales the function vertically, reducing its amplitude or range. The overall shape of the power function remains the same, but the function's maximum and minimum values are compressed towards the x-axis, while the horizontal properties, such as the domain and x-intercepts, remain unchanged.
Analyze the impact of vertical compression on the graph of an exponential function.
When an exponential function $f(x) = b^x$, where $b > 0$, undergoes vertical compression by a factor $a$, where $0 < a < 1$, the resulting function is $f(x) = a \cdot b^x$. This transformation scales the function vertically, reducing its amplitude or range, without affecting the function's horizontal properties, such as the base $b$ or the rate of growth. The compressed exponential function will still exhibit its characteristic exponential growth or decay pattern, but the overall height or vertical extent of the graph will be reduced.