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Chapter 2: Problem 134

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

Short Answer

Expert verified
To shrink a function's graph horizontally, multiply the 'x' values in the function equation by a factor greater than 1. This would result in a transformation from f(x) to f(bx) where b > 1.

Step by step solution

01

Understanding of function transformations

Functions can be shifted, reflected or 'shrunk'/'stretched'. This type of transformation is achieved by multiplying the 'x' or 'y' domain by a certain factor. If the 'x' value is multiplied by a factor greater than 1, a horizontal shrink happens. If the 'y' value is multiplied by a factor, it results in a vertical shrink/stretch.
02

Applying the Shrink Transformation

To shrink the graph of a function horizontally, adjust the 'x' term in the function. For a function f(x), this translates to a transformation to f(bx) where b > 1. This means, values of 'x' are multiplied by this factor 'b', causing the graph to shrink horizontally towards the y-axis. This is also known as a horizontal compression of the function.
03

Example

For example, consider the function f(x) = x^2. To shrink this function horizontally by a factor of 2, replace 'x' with '2x'. The new equation would be f(2x) = (2x)^2 = 4x^2. As you can see, all 'x' values are now half of their original distance from the y-axis, which means that the graph of the function has been shrunk horizontally.

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