Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Vertical Line Test to determine whether y is a function of x. Describe how you can use a graphing utility to produce the given graph. $$y=\frac{1}{2} x^{2}$$

Step-by-Step Solution

Answer

By applying the Vertical Line Test, it can be established that ‘y’ is indeed a function of ‘x’ as every vertical line drawn intersects the graph of the function at only one point. This means y is uniquely determined by x.

Step 1: Graph the Function

Firstly, input the function \(y=\frac{1}{2} x^{2}\) into the graphing utility. Graph this function, generating the shape of a parabola.

Step 2: Apply the Vertical Line Test

Draw a vertical line over the graph of the equation. If the line only touches the graph at one point no matter where it is drawn, then the equation is a function.

Step 3: Confirm the Analysis

Confirm from the drawn vertical lines that they only intersect the graph at one point irrespective of their position. Hence, 'y' is a function of 'x' because the Vertical Line Test is passed.

Key Concepts

Graphing Utility

A graphing utility is a valuable tool used to visualize mathematical equations and functions. It simplifies the process of plotting complex graphs, making it easier for you to understand various mathematical concepts visually. When you input a function into a graphing utility, it automatically generates a graph, saving time and reducing the likelihood of manual plotting errors.

To use a graphing utility effectively, follow these simple steps:

• Input the equation or function you wish to graph. For example, enter \(y=\frac{1}{2}x^2\).
• Select the graph type. In this case, you want a basic plot of the function.
• Adjust the viewing window to ensure the graph is visible and clear.
• Observe any patterns or intersections that occur on the graph.

By utilizing graphing utilities, you can easily explore and analyze mathematical relationships, which is particularly helpful for complex equations like quadratic functions.

Parabola

A parabola is the specific curve that results from the graph of a quadratic function such as \(y = \frac{1}{2}x^2\). It has a distinctive U-shape and is symmetric around its vertical axis, known as the axis of symmetry.

• The vertex is the highest or lowest point of the parabola, depending on its orientation.
• A parabola opens upwards if the coefficient of \(x^2\) is positive, and downwards if it is negative.
• The axis of symmetry for the parabola \(y = ax^2 + bx + c\) is the vertical line at \(x = -\frac{b}{2a}\).

In this exercise, the parabola described by \(y = \frac{1}{2}x^2\) opens upwards, indicating that the vertex is at the lowest point of the curve. Understanding the properties of parabolas helps you analyze their real-world applications, such as in physics and engineering scenarios.

Function

A function is a relationship between inputs and outputs where each input (often represented by \(x\)) corresponds to exactly one output (often represented by \(y\)). This relationship can be described using an equation or a graph.

The Vertical Line Test is a visual method used to determine if a relationship is a function. If a vertical line crosses the graph of an equation at only one point, then the equation represents a function.

With the equation \(y = \frac{1}{2}x^2\), the Vertical Line Test confirms it is a function because every vertical line drawn on its graph intersects the parabola at a single point. Understanding whether a given relationship is a function is fundamental in mathematics, as it ensures predictability and consistency in calculations.