Problem 30 - 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. \(k = 20\)

Step-by-Step Solution

Answer

The points \((1, 20), (2, 5), (3, 2.22), (4, 1.25)\) represent the inverse variation of \(y = \frac{20}{x^2}\)

Step 1: Understand the inverse variation model

The inverse variation model \(y = \frac{k}{x^2}\) means that \(y\) varies inversely with the square of \(x\) and \(k\) is the constant of variation. For this exercise, \(k = 20\). Let's choose some values for \(x\) and solve for \(y\).

Step 2: Substitute x values and solve

for instance, substitute \(x = 1, 2, 3, 4 \) in the equation \(y = \frac{k}{x^2}\) and solve for \(y\). For \(x = 1\), \(y = \frac{20}{1^2} = 20\), for \(x = 2\), \(y = \frac{20}{2^2} = 5\), for \(x = 3\), \(y = \frac{20}{3^2} \approx 2.22\), and for \(x = 4\), \(y = \frac{20}{4^2} = 1.25\). Hence, the points are \((1, 20), (2, 5), (3, 2.22), (4, 1.25)\)

Step 3: Plot the points on a rectangular coordinate system

Plot these points on a graph where the x-axis is for \(x\) and the y-axis is for \(y\). Doing this helps visualize the inverse relationship between \(x\) and \(y\) in the given equation.

Key Concepts

Rectangular Coordinate System

A rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane used for plotting points, lines, and curves. This system consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
Each point on this plane can be identified by a pair of coordinates \(x, y\), where \(x\) represents the position on the horizontal axis and \(y\) represents the position on the vertical axis.
In our exercise, the points derived from the inverse variation equation \(y = \frac{k}{x^2}\) are plotted on this system.

• The x-coordinate shows the independent variable, which is the value you choose.
• The y-coordinate results from computing the equation using the chosen x-value.

As you plot more points from these calculations, they form a curve that visually represents the relationship described by the equation.

Constant of Variation

In the inverse variation model, the term 'constant of variation' refers to \(k\) in the equation \(y = \frac{k}{x^2}\).
This constant is pivotal because it defines the scaling factor for the variation of y in relation to x.
In simpler language, \(k\) dictates how steep or flat the curve will be when plotted on the graph.

• For larger values of \(k\), the curve becomes steeper, as y decreases more quickly as x increases.
• For smaller values, the curve is flatter, indicating a slower change in y relative to changes in x.

In our specific problem, \(k = 20\). This means no matter the value of x, when you substitute it into the equation \(y = \frac{20}{x^2}\), the relationship stays defined by this constant value of 20. It's essential to remember that \(k\) does not change even as x and y pivot around the graph.

Plotting Points

Plotting points on a graph is the process of marking these ordered pairs on the rectangular coordinate system.
Each point can be visualized as a specific location that directly corresponds to values in our equation \(y = \frac{k}{x^2}\).
Here's how you plot these points:

• First, solve the equation for a chosen x-value to find the corresponding y-value.
• Next, write the results as ordered pairs, such as \(x, y\).
• Finally, locate these pairs on the graph by moving x units across the x-axis and y units up or down the y-axis, and draw a small dot at this point.

In our example, the points \(1, 20\), \(2, 5\), \(3, 2.22\), and \(4, 1.25\) were plotted.
This graph clearly displays the "inverse" relationship: as the x-value gets larger, the corresponding y-value gets smaller. Once plotted, these points illustrate a curve that helps visualize the functional relationship between x and y when described by inverse variation.