Problem 28 - Precalculus with Limits

Question

In Exercises 27-30, use the given value of \(k\) to complete the table for the inverse variation model \(y = \frac{k}{x^2}\) Plot the points on a rectangular coordinate system. \(k = 5\)

Step-by-Step Solution

Answer

The graph will be a curve showing the inverse variation between \(x\) and \(y\). The y-values decrease as the x-values increase, demonstrating the nature of inverse variation. The exact values and shape of the curve will depend on the chosen \(x\) values for the table.

Step 1: Identify variable and constant values

The provided equation is \(y = \frac{k}{x^2}\). Here, \(k\) is the constant and its given value is 5. \(x\) is the variable and changes. Given ranges of \(x\) we will solve for \(y\).

Step 2: Substitute \(k\) into the equation

Substitute the given value of \(k=5\) into the equation, resulting in \(y = \frac{5}{x^2}\).

Step 3: Evaluate \(y\) for various \(x\) values

Choose a set of \(x\) values, substitute each into the equation, and solve for \(y\). For example, if \(x=1\), you would get \(y = \frac{5}{1^2} = 5\). Repeat this calculation for other \(x\) values.

Step 4: Create the table and plot the points on a graph

You should now have a collection of \((x,y)\) pairs. Create a table using these values and then plot these points on a rectangular coordinate system. The graph will represent the inverse variation.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane used to visualize and interpret mathematical relationships between variables. In this system, each point is uniquely defined by a pair of numerical coordinates. These coordinates correspond to the distances from two fixed perpendicular lines called axes.

The horizontal axis is known as the x-axis and the vertical axis is the y-axis. Where these two axes intersect is called the origin, typically represented by the coordinates (0,0). This framework provides a simple yet powerful way to depict complex mathematical equations like inverse variation models. It assists in plotting points, allowing us to see relationships visually.

Plotting Points

Plotting points is a fundamental step in graphing any equation in the rectangular coordinate system. Each point on a plane is represented as an ordered pair \(x, y\), where \(x\) is the position along the x-axis, and \(y\) corresponds to the y-axis.

When plotting, you start at the origin (0,0). From there, move horizontally to the specified x-coordinate. Then, move vertically to the given y-coordinate to mark the point.

Let's consider the equation \(y = \frac{5}{x^2}\). To plot points for various values of \(x\):

• If \(x = 1\), then \(y = 5\). The point is \( (1,5) \).
• If \(x = 2\), then \(y = \frac{5}{4} = 1.25\). The point is approximately \( (2,1.25) \).

By repeating this process for multiple \(x\) values, you can generate a series of points to be plotted, eventually tracing the curve that represents the equation.

Mathematical Modeling

Mathematical modeling involves using mathematical equations to represent real-world phenomena and relationships. Inverse variation, expressed as \(y = \frac{k}{x^2}\), is a type of mathematical model. It signifies that one quantity increases as another decreases, and vice versa.

In the original exercise, we have \(k = 5\). This constant helps determine the behavior of the model. As \(x\) increases, \(y\) will decrease because \(x^2\) appears in the denominator. This model is useful for scenarios where there is an inverse relationship between variables, like gravity's influence diminishing with distance.

Mathematical models are critical because they allow us to predict and understand variable interactions and their impacts on each other. This predictive ability is invaluable in fields such as physics, economics, and biology.

Function Evaluation

Function evaluation is the process of finding the output value \(y\) for a given input value \(x\) in a function. For the equation \(y = \frac{5}{x^2}\), function evaluation helps us determine the y-values that correspond to different x-inputs.

To evaluate a function for specific x-values:

• Substitute the x-value into the function. For example, substituting \(x = 1\) results in \(y = \frac{5}{1^2} = 5\).
• Use arithmetic operations to simplify and solve for \(y\).
• Repeat for multiple values of \(x\) to observe how \(y\) changes and to gather data for graphing.

This process allows us to build a table of values, showing how changes in \(x\) affect \(y\), which is crucial for understanding the behavior of the inverse variation function and representing it graphically.