Problem 64 - 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. $$ y=\sqrt{4-x^{2}}, \text { compressed vertically by a factor of } 3 $$

Step-by-Step Solution

Answer

The new equation is \( y = \frac{1}{3}\sqrt{4-x^2} \).

Step 1: Understanding Vertical Compression

Vertical compression of a graph implies multiplying the function's output by a value less than 1. In this case, we need to compress the graph vertically by a factor of 3, which translates to the multiplication of the entire function by \( \frac{1}{3} \).

Step 2: Formulate the New Equation

Given the original function \( y = \sqrt{4-x^2} \), apply the vertical compression factor: \( y = \frac{1}{3}\sqrt{4-x^2} \). This creates a new graph that is vertically compressed by the specified factor of 3.

Key Concepts

Transformation of Functions

Transformations of functions involve changing the appearance of the graph without altering the essential nature of the function. These transformations can include shifting, reflecting, stretching, or compressing the graph in various directions. When you transform a function, you are applying rules that either change the position, size, or shape of the graph.

A key aspect of transforming functions is understanding how operations modify a function's output or input. For instance, if you multiply the output by a constant, you're either stretching or compressing the graph vertically. Similarly, altering the input can lead to horizontal changes.

Transformations are a powerful tool in understanding and visualizing functions better. They help in graphing complex equations more easily by starting from a basic form and applying transformations as necessary.

Graphical Transformations

Graphical transformations involve manipulating the graph of a function to achieve particular changes in its appearance. These changes can be broadly categorized into:

• Vertical Transformations: These affect the up-or-down appearance of the graph. Multiplying the function by a constant less than 1 compresses it vertically, while a constant greater than 1 stretches it.


• Horizontal Transformations: Involve changes along the x-axis. These adjustments happen when you alter the function's input, often resulting in shifts or stretches sideways.


• Reflections: Flipping the graph over a particular axis, usually by introducing a negative sign either inside or outside the function.

When considering vertical compression specifically, you're focusing on reducing the height of each point on the graph without altering its horizontal location. This is what happens when you multiply the output of the function by a fraction smaller than 1.

Square Root Functions

Square root functions, like \( y = \sqrt{x} \), have a unique, characteristic shape. They start at a certain point, usually on the positive x-axis, and curve upwards gently. The basic square root function is important because many transformations build off this initial shape.

Applying transformations, like compressing or stretching, alters how steeply or flatly this curve rises. For example, if you compress the square root function vertically by a factor of \( \frac{1}{3} \), each point is brought closer to the x-axis, which makes the curve flatter. The compression does not change the starting point (origin), nor does it affect how far along the x-axis the graph stretches.

Understanding the square root function is essential as it's a foundation for more complex functions. When you are tasked with applying a transformation such as vertical compression, it helps to visualize how each point on \( \sqrt{x} \) behaves under the transformation rules, ensuring accurate graphing and comprehension.