Problem 27 - 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. \(k = 2\)

Step-by-Step Solution

Answer

For the selected x-values: -3, -2, -1, 1, 2, and 3, the corresponding y-values as calculated from the inverse variation equation \( y = \frac{2}{x^2} \) are roughly 0.222, 0.5, 2, 2, 0.5, and 0.222 respectively. After plotting these points on a rectangular coordinate system, a curved line appears from the top right, passing by the top and bottom right quadrants and extending to the top left quadrant.

Step 1: Identify the X-values

First, select distinct but reasonable x-values to use. On a typical graph, these could be -3, -2, -1, 1, 2, 3 for instance.

Step 2: Calculate the Y-values

For each selected x-value, substitute it into the inverse variation equation \( y = \frac{2}{x^2} \) to calculate the corresponding y-values.

Step 3: Complete the Table

At this point, you will have a set of (x, y) pairs which satisfy the given equation. Write down these pairs in the table.

Step 4: Plot the Points

Finally, plot each point on a rectangular coordinate system. Remember, the x-value determines the distance left or right from the origin, while the y-value determines the distance up or down from the origin. For negative x-values, move to the left of the origin; for positive x-values, move to the right. For negative y-values, move downwards from the origin; for positive y-values, move upwards.

Key Concepts

Rectangular Coordinate System

Understanding the rectangular coordinate system is essential when graphing equations like the inverse variation model. This system is made up of two perpendicular lines, known as the x-axis (horizontal) and the y-axis (vertical).
These axes intersect at a point called the origin, which has coordinates (0, 0). The coordinate system is divided into four quadrants, with each quadrant represented by a combination of positive and negative x and y values.

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, y is negative.

When plotting points, the x-value indicates the horizontal distance from the origin, and the y-value indicates vertical distance. Moving along these axes helps to locate precise points that represent solutions to equations on a graph.

Inverse Variation Model

The inverse variation model describes a specific mathematical relationship between two variables. In this context, when one variable increases, the other decreases. It is expressed by the equation \[ y = \frac{k}{x^2} \]Here, \( y \) and \( x \) are the variables, and \( k \) is a constant. The inverse relationship is evident because as \( x \) increases, \( y \) decreases due to the division, and vice versa.
This type of equation is often used to model real-world situations where one quantity inversely affects another, such as speed and travel time. Identifying the constant \( k \) is crucial, as it scales the relationship and determines how steep the graph will be.

Plotting Points

Plotting points on a rectangular coordinate system based on the inverse variation model involves determining specific x-values and then using the equation to calculate the corresponding y-values.
For example, with the equation \( y = \frac{2}{x^2} \), and choosing valuation like -3, -2, -1, 1, 2, 3, you calculate \( y \) for each \( x \) and get a set of ordered pairs like (-3, y), (-2, y), etc.
Once the pairs are obtained, you can place them precisely on the graph by starting at the origin and moving along the x-axis and y-axis according to the values obtained:

• Positive x-values move right from the origin; negative x-values move left.
• Positive y-values move upwards; negative y-values move downwards.

This plotting process allows you to visualize the inverse relationship, helping solidify the understanding of how variables interact through this model in a tangible graph form.