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Chapter 2: Problem 133

What must be done to a function's equation so that its graph is stretched vertically?

Short Answer

Expert verified
To perform a vertical stretch of a function's graph, multiply the entire function by a constant factor, \( a \), where \( a > 1 \), which gives a vertical stretch, and \( 0 < a < 1 \) gives a vertical shrink. The resulting equation is \( y = a*f(x) \).

Step by step solution

01

Understanding the concept of vertical stretch

When we talk about stretching a function's graph vertically, we're referring to increasing or decreasing the y-values of the function by a certain factor. This results in the graph appearing 'taller' or 'shorter' when stretched.
02

Applying Vertical Stretch to A Function

In order to apply a vertical stretch to a function's equation, say \( y = f(x) \), we multiply the entire function by a constant factor, \( a \). If \( a > 1 \), we get a vertical stretch, and if \( 0 < a < 1 \), we get a vertical shrink. The new function would be \( y = a*f(x) \). The factor \( a \), where \( a > 0 \), determines how much stretching or shrinking is applied. The larger the value of \( a \), the more the function is stretched vertically.
03

Illustration of Vertical Stretch

As an example, consider the function \( y = sin(x) \). The graph of \( y = 2sin(x) \), is a vertical stretch of the initial sin function by a factor of 2. This means that all the sinusoidal peaks, which were initially at 1, are now located at 2, and likewise troughs have gone from -1 to -2. In other words, the graph is 'taller' or 'stretched'.

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