Question

The graph of the function \(f\) is to be transformed as described. Find the function for the transformed graph. \(f(x)=\sqrt{x^{2}+4}\); compressed vertically by a factor of 2

Short answer

The function for the transformed graph is \(g(x) = \frac{\sqrt{x^2 + 4}}{2}\).

Step-by-step solution

Identify the given function

The given function is \(f(x) = \sqrt{x^2 + 4}\).

Apply the vertical compression

To apply the vertical compression, we need to halve the value of the function for every x, i.e., g(x) = f(x)/2. So, g(x) = (\(f(x)\))/2.

Substitute the given function for f(x)

We substitute f(x) = \(\sqrt{x^2 + 4}\) in g(x) to get the transformed function: g(x) = (\(\sqrt{x^2 + 4}\))/2.

Write the final answer

The function for the transformed graph is \(g(x) = \frac{\sqrt{x^2 + 4}}{2}\).

Key concepts

Vertical Compression

When we talk about vertical compression in the context of function transformations, we're referring to a specific type of modification to the function's graph. This transformation affects the way the graph is stretched or squeezed along the y-axis.

To visualize a vertical compression, imagine the y-values of a function's graph being pushed closer together, effectively reducing their range. Mathematically, this is achieved by multiplying the function's output by a factor between 0 and 1. If the original function is described by f(x), then after a vertical compression by a factor of k, the new function g(x) will be defined as g(x) = k \times f(x), with 0 < k < 1.

For instance, the given exercise required a vertical compression by a factor of 2, meaning each output value of the original function f(x) = \( \(sqrt{x^2 + 4}\) \) is halved. Hence, the transformed function is g(x) = \( \(frac{\sqrt{x^2 + 4}}{2}\) \).

Square Root Functions

A square root function is a type of radical function where the primary operation is finding the square root of the input variable. The general form is f(x) = \( \(sqrt{x}\) \). These functions are essential in mathematics as they model various real-world scenarios which require finding the root of a quantity.

A square root function typically has a graph that is a smooth curve, starting at the origin (0,0), and increasing at a decreasing rate as x gets larger. These functions only yield real number outputs for nonnegative input values, which means their domain is x ≥ 0. However, when the square root function includes an added constant or is part of a more complex expression—as in the exercise f(x) = \( \(sqrt{x^2 + 4}\) \)—the domain and the shape of the graph can alter accordingly.

Transformation of Functions

The transformation of functions is a fundamental concept in algebra and precalculus that involves altering the appearance of a graph in various ways without changing the input-output relationship of the original function. The primary types of transformations include translations (shifts), reflections, stretches, and compressions.

Translations move the graph up, down, left, or right, while reflections flip it over the x-axis or y-axis. Stretches and compressions change the graph's size either horizontally or vertically. When we apply a transformation to a function f(x), we modify it to a new function g(x) that maintains the basic shape but is repositioned or resized.

In our exercise, the original square root function undergoes a vertical compression, which is a type of transformation that maintains the function's general shape while making it 'narrower' along the y-axis.

Radical Functions

Radical functions are a category of functions that involve roots, such as square roots, cube roots, etc. The most common type of radical function is the square root function, but radical functions can also involve higher roots. The general form is f(x) = \( \sqrt[n]{g(x)}\), where \sqrt[n]{} denotes the n-th root and g(x) is some function of x.

These functions are important for solving equations where the variable is under a radical. The properties of radical functions, such as their domain and range, depend on the degree of the root and the function within the radical. For example, the function from our exercise f(x) = \( \(sqrt{x^2 + 4}\) \) involves a square root of a quadratic expression, which significantly differs from a basic square root function in terms of its graph and properties.