Problem 55 - Thomas Calculus

Question

Exercises \(51-60\) 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 function is compressed horizontally by a factor of 4 to become \( y = \sqrt{4x+4} \).

Step 1: Identify the Transformation Type

The problem requires a horizontal compression of the function by a factor of 4. In transformations, horizontal stretching or compressing affected by a factor, say \( a \), is done inside the function as \( \sqrt{x} \to \sqrt{ax} \). For compression by a factor of 4, we use \( a = 4 \).

Step 2: Apply the Horizontal Compression Transformation

Since we need to compress horizontally by a factor of 4, change the variable inside the square root from \( x+1 \) to \( 4(x+1) \). This means the function \( y = \sqrt{x+1} \) becomes \( y = \sqrt{4(x+1)} \).

Step 3: Simplify the New Function Equation

Distribute the 4 to \( x+1 \) to get \( 4x + 4 \). Thus, the transformed function is \( y = \sqrt{4x+4} \).

Key Concepts

Horizontal Compression

Horizontal compression is a graph transformation that changes how a function appears along the x-axis. When you compress a graph horizontally, you make it "squeeze" closer together. This affects the width of the graph without altering its vertical height. If we need to compress a graph horizontally by a factor of 4, it means we replace each x-value of the function with one-fourth of that value.

Understanding the Process:

• The transformation involves changing the input (x-values).
• For a function like \( y = f(x) \), horizontal compression by a factor of \( a \) is represented as \( y = f(ax) \).
• In our example, we worked with \( y = \sqrt{x+1} \) and applied a horizontal compression by introducing a factor 4 inside the square root: \( y = \sqrt{4(x+1)} \).

In this transformation, every x-value is effectively multiplied by 4 inside the function, thereby achieving the horizontal compression of the graph. Think of it as "speeding up" the graph's progression across the x-axis.

Square Root Function

A square root function is one of the fundamental types of functions in mathematics, taking the form \( y = \sqrt{x} \). The square root function has specific characteristics:

• It exists only for non-negative values of x (i.e., \( x \geq 0 \)), since you cannot take the square root of a negative number in the set of real numbers.
• The basic shape of its graph is indicative of a gentle, gradual rise from left to right, starting at the origin (0,0) and extending indefinitely.
• It grows slower and reacts less dramatically to changes in x compared to linear functions.

For our exercise, \( y = \sqrt{x+1} \), the function is slightly shifted horizontally to accommodate the \( +1 \) inside the root. It means the "beginning" of the curve moves left by 1 unit on the graph. When transforming this function including a horizontal compression, as in \( y = \sqrt{4(x+1)} \), it changes the pace at which the graph rises as you move along the x-axis.

Graph Transformations

Graph transformations are operations that alter the appearance or position of a graph in the coordinate plane. When we apply transformations to a function, we can stretch, compress, shift, or reflect it. This helps in understanding how functions behave under different changes.

Here is a breakdown of basic transformations applied to functions:

• Horizontal Shifts: Moving the graph left or right by adding or subtracting a value inside the function. For instance, \( y = \sqrt{x+1} \) shifts left by 1 unit compared to \( y = \sqrt{x} \).
• Vertical Shifts: Moving the graph up or down by adding or subtracting from the function. For example, \( y = \sqrt{x} + 2 \) shifts the graph upwards by 2 units.
• Horizontal Compression/Stretch: Affects the x-values in the function. In a compression by a factor \( a \), the x-values are "sped up" by replacing \( x \) with \( ax \).
• Vertical Compression/Stretch: Similar to horizontal but affects y-values by multiplying the whole function.

These transformations combine to form the new equation and produce the desired look of the original function on the graph. They let you manipulate the function to fit specific criteria without altering its inherent characteristics.