Problem 29 - 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. \(k = 10\)

Step-by-Step Solution

Answer

The inverse variation model equation \(y = \frac{k}{x^2}\), where \(k = 10\) is used to generate a set of ordered pairs \((x, y)\). These pairs can be plotted on a rectangular coordinate system to observe the relationship inline with inverse variation model.

Step 1: Understand the inverse variation model equation

Recognize that in the equation \(y = \frac{k}{x^2}\) model, where \(k\) is a constant. The goal is to find corresponding values of \(y\) for various \(x\) values with a given \(k\).

Step 2: Substituting 'k' and computing 'y' values

Given \(k = 10\) and various \(x\) values (say \(x = 1,2,3,4,5\)), substitute these values into the equation to get corresponding \(y\) values. Use the formula \(y = \frac{10}{x^2}\) to compute the values of \(y\).

Step 3: Plotting on the coordinate system

Once you have the \(x\) and \(y\) values from Step 2, you can plot these ordered pairs \((x, y)\) on a rectangular coordinate system. Observe the pattern or shape formed by the plot.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian plane, helps us visualize relationships between variables. It consists of two perpendicular axes: the horizontal axis, called the x-axis, and the vertical axis, called the y-axis. Their intersection is the origin, labeled as \(0,0\). Each point on this plane is represented as an ordered pair \( (x, y) \).

• The x-coordinate tells us how far along the horizontal axis a point is.
• The y-coordinate shows the position along the vertical axis.

Using this system, we can graph equations like \(y = \frac{k}{x^2}\), helping us see patterns in the data. By plotting points found in a table, we can understand the inverse variation visually.To start, simply mark the points on the grid and look for any shape or trend they form.

Equation Substitution

Equation substitution is a crucial technique in algebra, involving replacing variables with specific values. In the equation \(y = \frac{k}{x^2}\), we can substitute a given constant \(k = 10\) and various values of \(x\) to determine \(y\).For example:

• When \(x = 1\), \(y = \frac{10}{1^2} = 10\)
• When \(x = 2\), \(y = \frac{10}{2^2} = 2.5\)
• Continue this process for other values of \(x\).

Substituting different values of \(x\) gives us a complete set of ordered pairs \( (x, y) \) that we can plot. This simple method allows us to transform an abstract equation into concrete data points.

Mathematical Modeling

Mathematical modeling involves creating simplified representations of real-world phenomena using equations and formulas. In this exercise, the inverse variation model \(y = \frac{k}{x^2}\) shows how \(y\) changes as \(x\) changes.
This model illustrates a specific relationship where \(y\) decreases as \(x\) increases, provided \(k\) is constant. In practical applications, this can represent numerous situations, such as light intensity varying with distance or certain economic behaviors.

• A model helps predict behavior without conducting experiments.
• We can adjust models to fit different scenarios by altering parameters like \(k\).

Mathematically modeling relationships teaches us to interpret and describe real-world behaviors efficiently.

Plotting Graphs

Plotting graphs is a method to visually express mathematical relationships. After calculating the \(y\) values for each \(x\), we can use these ordered pairs to create a graph.Start by marking each \( (x, y) \) pair on the rectangular coordinate system. Connect the points smoothly, if possible, to reveal the pattern of the graph. With inverse variation, you'll notice a decreasing curve.

• This curve shows how \(y\) diminishes as \(x\) increases.
• Analyze the curve’s shape to understand the inverse relationship.

Graphing provides a clear picture of how variables relate in a way that numbers alone may not convey. This visual comprehension can make complex relationships more straightforward and intuitive.