Problem 53 - 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 . $$ y=1+\frac{1}{x^{2}}, \quad \text { compressed vertically by a factor of } 2 $$

Step-by-Step Solution

Answer

New equation: \( y = \frac{1}{2} + \frac{1}{2x^2} \).

Step 1: Understand the Transformation

We are asked to perform a vertical compression by a factor of 2 on the function given by \[ y = 1 + \frac{1}{x^2}. \] This means that every y-value of the function will be halved, effectively reducing the height of the graph by 50%.

Step 2: Apply the Vertical Compression

To apply a vertical compression by a factor of 2, every y-value, for each corresponding x, should be divided by 2. Therefore, the equation becomes:\[ y = \frac{1}{2}(1 + \frac{1}{x^2}) = \frac{1}{2} + \frac{1}{2x^2}. \]

Step 3: Rewrite the Equation

After applying the transformation, the new equation of the function after compression is:\[ y = \frac{1}{2} + \frac{1}{2x^2}. \] This is the equation of the graph that is vertically compressed by a factor of 2.

Key Concepts

Vertical Compression

A vertical compression is a type of function transformation that alters the graph of a function by bringing it closer to the x-axis. This transformation affects the y-values of the functions without modifying the x-values. In simpler terms, the height of the graph is reduced. To perform a vertical compression, we multiply all the y-values of the function by a constant factor less than 1.

Let's explore using our example function:

• Original function: \( y = 1 + \frac{1}{x^2} \)
• Compression factor: 2

To vertically compress the graph by a factor of 2, you divide the entire function by 2. This results in the new function: \( y = \frac{1}{2} + \frac{1}{2x^2} \). Notice how each term of the original function is divided by 2 to achieve the compression. This effectively reduces every y-value of the graph by half, making the graph appear squeezed vertically.

Function Transformation

Function transformations include several methods of modifying a function's graph, such as translations, reflections, stretches, and compressions. These transformations affect the position, size, or orientation of the graph but keep the underlying function the same.

Vertical and horizontal transformations are the most common, often visualized by how they change the graph's shape or location in the coordinate plane.

• Vertical transformations affect the output (y-value), directly modifying the graph's height.
• Horizontal transformations alter the input (x-value), affecting the graph's width or position along the x-axis.

In our exercise, we focused on a vertical transformation, namely vertical compression. This type of transformation changes the scale of the graph without affecting its basic shape or orientation. Each y-value is replaced by a fraction of its original value, depending on the compression factor, thereby making the overall graph appear smaller vertically.

Algebraic Manipulation

Algebraic manipulation is a critical skill for transforming functions and simplifying expressions. It involves rearranging and rewriting expressions to make them easier to work with or to achieve a specific form. This process is fundamental when performing graph transformations.

For the function \( y = 1 + \frac{1}{x^2} \), algebraic manipulation allows us to correctly apply the vertical compression:

• First, recognize that all terms need to be adjusted. Here, both the constant and the fraction are reduced by the compression factor.
• Next, rewrite the function as: \( y = \frac{1}{2}(1) + \frac{1}{2}(\frac{1}{x^2}) \)

This step-by-step breakdown highlights the importance of applying transformations evenly across all parts of the function. By manipulating the algebraic equation correctly, we accurately reflect the intended graph transformation in the new function form, thus preserving the graph's mathematical integrity.