Problem 134 - College Algebra

Question

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

Step-by-Step Solution

Answer

To shrink a function's graph horizontally, you need to multiply the x-values (or the independent variable) in the function's equation by a factor greater than 1.

Step 1: Understand the Term 'Horizontal Shrinkage'

In the graph of a function, a 'horizontal shrinkage' refers to the transformation where the graph appears to be squeezed towards the y-axis. It occurs when every x-coordinate of the original function gets multiplied by a factor greater than 1.

Step 2: Describing the Transformation

To shrink the graph of a function horizontally, the x-values in the function's equation need to be multiplied by a factor greater than 1. So from a function \(f(x)\), if the x-values are replaced with \(kx\), where \(k > 1\), the function's graph will be shrunk horizontally towards the y-axis. This is denoted by \(f(kx)\), where \(k\) is the factor by which x is being multiplied.

Step 3: Illustrate with an Example

Let's take the function \(f(x) = x^2\). To shrink this graph horizontally, the function becomes \(f(kx) = (kx)^2\). If \(k = 2\), then the function will be \(f(2x) = (2x)^2\), which indicates that the graph of the function will be horizontally shrunk by half.