Problem 61 - Thomas Calculus

Question

Exercises \(57-66\) tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph. $$ y=\sqrt{x+1}, \quad \text { compressed horizontally by a factor of } 4 $$

Step-by-Step Solution

Answer

The equation is \( y = \sqrt{4x + 1} \).

Step 1: Identify the Transformation

First, note that the function given is \( y = \sqrt{x+1} \). The problem states that this function is to be compressed horizontally by a factor of 4.

Step 2: Apply Horizontal Compression

To compress a function horizontally by a factor of \( c \), replace \( x \) with \( cx \) in the equation. For a compression factor of 4, we substitute \( x \) with \( 4x \) in the function \( \sqrt{x+1} \).

Step 3: Write the Transformed Equation

The equation for the horizontally compressed function is then \( y = \sqrt{4x + 1} \). This results in the graph being compressed horizontally by a factor of 4.

Key Concepts

Horizontal Compression

Horizontal compression is a type of transformation applied to functions that affects the width of their graphs on the coordinate plane. To achieve horizontal compression, we alter the input variable, typically represented by "x," in the function's equation.

Consider the function transformation process: in the original function, you replace every occurrence of "x" with "cx," where "c" is the compression factor.

• If "c" is greater than 1, the graph compresses horizontally. This makes the graph squish together horizontally, effectively reducing its width.
• This type of transformation makes the function appear to "speed up" along the x-axis.

In our example, with a horizontal compression of factor 4, the variable "x" in the function gets replaced with "4x." This results in the horizontally compressed function equation:
\[ y = \sqrt{4x + 1} \]
The transformation changes the relationship between x and y, making any changes along the x-axis affect the output more significantly.

Graph Transformations

Graph transformations encompass a variety of changes you can make to the graph of a function. These include translations, stretches, compressions, and reflections, which alter how the graph appears visually without fundamentally changing its overall form.

• Horizontal Compression or Stretch: As discussed, changing "x" to "cx" compresses (if "c" > 1) or stretches (if "c" < 1) horizontally.
• Vertical Compression or Stretch: In contrast, modifying the entire function (e.g., multiplying the function by a factor "c") affects its height.
• Translations: We can shift graphs up, down, left, or right by adjusting the function’s constants.
• Reflections: Flipping the graph over an axis involves changing the sign of the function or its variable.

Understanding these transformations is crucial to graph analysis and function manipulation. They allow you to predict how a mathematical equation will look visually, and how changes to the function's equation affect its overall graph. In our case, a horizontal compression by a factor of 4 transforms \( y = \sqrt{x+1} \) into \( y = \sqrt{4x+1} \). This modifies the width of the square root function's graph, compressing it horizontally.

Square Root Functions

Square root functions are a type of radical function that include the square root of "x" or expressions involving "x." They take the general form \( y = \sqrt{x} \), and they feature a distinct shape, known as a "half-parabola," which originates at the point where the expression inside the square root becomes zero.

Key properties of square root functions:

• The domain of a basic square root function \( y = \sqrt{x} \) is \( x \geq 0 \), since square roots of negative numbers are not defined within the real number system.
• The range also starts at 0 and extends to infinity: \( y \geq 0 \).
• They increase at a decreasing rate: as "x" becomes larger, the rate of increase slows.

In our specific context, the function \( y = \sqrt{4x+1} \) results from applying a horizontal compression to the function, meaning that for a given output "y," you need a larger input "x". This alteration compresses the graph along the x-axis but maintains the unique shape typical of square root functions.