Problem 59 - Algebra and Trigonometry

Question

If \(f(x)=\sqrt{2 x-x^{2}}\) graph the following functions in the viewing rectangle \([-5,5]\) by \([-4,4]\) , How is each graph related to the graph in part (a)? \(\begin{array}{llll}{\text { (a) } y=f(x)} & {\text { (b) } y=f(2 x)} & {\text { (c) } y=f\left(\frac{1}{2} x\right)}\end{array}\)

Step-by-Step Solution

Answer

Graph (b) is a horizontal compression and graph (c) is a horizontal stretch compared to graph (a).

Step 1: Understanding the Parent Function

The parent function is given as \( y = f(x) = \sqrt{2x - x^2} \). This defines a semi-circle centered at the origin and covers the domain where \(2x - x^2 \geq 0\). By solving \(x(2-x) \geq 0\), we find the domain to be \([0, 2]\).

Step 2: Graph Part (a) - y=f(x)

Within the domain \([0, 2]\), plot the graph of \(y = f(x) = \sqrt{2x - x^2}\) within the viewing rectangle \([-5,5] \text{ by } [-4,4]\). It will appear as a semi-circle with radius 1 centered at point (1,0).

Step 3: Graph Part (b) - y=f(2x)

For the transformation \(y = f(2x)\), substitute \(2x\) into the function: \(y = \sqrt{2 \cdot (2x) - (2x)^2} = \sqrt{4x - 4x^2}\). The domain of \( \sqrt{4x - 4x^2} \) is \([0, 1]\), indicating a horizontal compression. Graph this within the viewing rectangle. The width shrinks by half compared to the parent function.

Step 4: Graph Part (c) - y=f(1/2 x)

For \(y = f(\frac{1}{2}x)\), substitute \(\frac{1}{2}x\) into the function: \(y = \sqrt{2 \cdot \left(\frac{1}{2}x\right) - \left(\frac{1}{2}x\right)^2} = \sqrt{x - \frac{1}{4}x^2}\). The domain of this function is \([0, 4]\). This graph is a horizontal stretch, twice as wide as the parent function. Plot this in the rectangle \([-5,5] \text{ by } [-4,4]\).

Step 5: Analyze Graph Relationships

Graph (b) is a horizontal compression by factor 1/2 compared to graph (a). Graph (c) is a horizontal stretch by a factor of 2 compared to graph (a). These transformations are impacts of coefficients inside the function argument \(f\).

Key Concepts

Understanding Domain and Range

When talking about graphs of functions, there are two essential terms to understand: domain and range. Think of domain as all the possible values you can plug into a function. For the parent function given, \(f(x) = \sqrt{2x - x^2}\), the domain is found where the function under the square root is non-negative. To determine this, solve \(2x - x^2 \geq 0\). The values of \(x\) that satisfy this inequality are in the interval \([0, 2]\). This means you can input any \(x\) between 0 and 2 into the function.

On the other hand, range refers to all the possible outputs of the function. After finding the domain, you can substitute these values back into the function to determine the outputs. Because of the square root nature of the function, the outputs (or range) will be non-negative. For \(f(x) = \sqrt{2x - x^2}\), as \(x\) ranges from 0 to 2, the maximum point is at \(x = 1\), giving a maximum value output of 1. Hence, the range is from 0 to 1, written as \([0, 1]\).

Knowing the domain and range is crucial as it tells us where the function is defined and what the possible outputs are.

Delving into Function Transformations

Function transformations are changes made to a graph that affect its shape, position, or size. There are several types of transformations that can occur: horizontal shifts, vertical shifts, stretching, and compression. These transformations can be tricky, but understanding them is essential.

Let's examine the two transformations in the problem:

• For \(y = f(2x)\), substitute into the parent function to obtain \(y = \sqrt{4x - 4x^2}\). Compared to \(f(x)\), this is a horizontal compression by a factor of 1/2. Essentially, the graph becomes narrower, compressing the domain to \([0, 1]\).
• For \(y = f(\frac{1}{2}x)\), substitute to get \(y = \sqrt{x - \frac{1}{4}x^2}\). This is a horizontal stretch by a factor of 2, essentially widening the graph, expanding the domain to \([0, 4]\).

Horizontal stretches and compressions affect only the \(x\)-value inputs, broadening or narrowing the graph's appearance.

These transformations help you see how changing the inputs of a function (the \(x\) values) affects its graphical representation.

Insights into Parent Function Analysis

A parent function is the simplest form of a set of functions that remain in a family or category. In our exercise, the parent function is \(y = f(x) = \sqrt{2x - x^2}\), a specific semi-circle shape. Understanding this basic shape helps in anticipating what happens after transformations.

In this example, the parent function forms a semi-circle spanning from \(x=0\) to \(x=2\) and has a height reaching from \(y=0\) to \(y=1\). This gives you a baseline for comparison. Knowing the parent function means:

• You can predict outcomes when adjustments are made, like those in the transformations \(f(2x)\) and \(f(\frac{1}{2}x)\).
• You can determine the exact changes in the shape of the graph concerning the x-axis.

Understanding the parent function allows you to have a reference point when graphing any given function, making analysis easier and more intuitive with transformations.