Problem 34 - Algebra 1

Question

Make a table of values for x = 1, 2, 3, and 4. Use the table to sketch a graph. Decide whether x and y vary directly or inversely. $$y=\frac{3}{2 x}$$

Step-by-Step Solution

Answer

The table of values yields \((x, y)\) coordinates of \((1, 1.5), (2, 0.75), (3, 0.5), (4, 0.375)\). After sketching the graph, it can be concluded that \(x\) and \(y\) are inversely proportional because \(y\) decreases as \(x\) increases.

Step 1: Create a Table of Values

Take the given equation \(y = \frac{3}{2x}\) and plug in the values of x as 1, 2, 3, and 4 to find the corresponding y-values. This will create a table of \(x, y)\) coordinates.

Step 2: Sketch the Graph

Plot the points from the table on a coordinate system and connect the points to get the graph of the equation \(y = \frac{3}{2x}\).

Step 3: Determine the Relationship

Examine the graph and the equation. If y decreases as \(x\) increases, and the equation takes the form \(y = \frac{k}{x}\), where \(k\) is a constant (in this case 3/2), then \(x\) and \(y\) are inversely proportional. If \(y\) increased as \(x\) increases, they would be directly proportional. Based on the graph and the equation, \(x\) and \(y\) will be found to be inversely proportional.

Key Concepts

Tables of Values

Creating a table of values is an effective way to visualize the relationship between variables in an equation. In the exercise with the equation \(y = \frac{3}{2x}\), you want to find the corresponding \(y\) values for different \(x\) values. - Start by selecting a set of \(x\) values. Common choices are 1, 2, 3, and 4, as these are easy to plug into the equation and calculate. - For each \(x\), substitute it into the equation to solve for \(y\). For example, when \(x = 1\), the calculation is \(y = \frac{3}{2 \cdot 1} = \frac{3}{2}\). Repeat this process for \(x = 2\), \(x = 3\), and \(x = 4\) to fill out the table. This creates a simple list of pairs (\(x, y)\) that clearly represent how one variable affects the other. Once the table is ready, it becomes a roadmap for graphing the relationship.

Graphing Equations

Graphing an equation helps in understanding visually how the variables are related. With the table of values from our previous step, you can now plot these points onto a graph. Each point \((x, y)\) from your table corresponds to a location on the graph. - Start by marking each point on the graph according to its coordinates. - Once all points are marked, connect them smoothly to illustrate the trend or pattern they form. In this particular exercise, the graph of \(y = \frac{3}{2x}\) will not be a straight line but rather a curve. As \(x\) increases, \(y\) decreases, creating a downward trending curve as you move from left to right.This visual representation helps to clearly show whether there are direct or inverse relationships, paving the way for further analysis.

Coordinate Systems

Understanding the coordinate system is crucial when plotting points from an equation. The graph lies on a two-dimensional plane, separated by two perpendicular lines called the axes. - The horizontal axis is known as the \(x\)-axis. It lists the possible values of \(x\).- The vertical axis is known as the \(y\)-axis, showing values for \(y\). Every point on this plane can be described by a coordinate pair \((x, y)\). When you plot the pairs from your table, you're essentially drawing a point at the intersection of the corresponding \(x\) and \(y\) values on these axes. In the case of \(y = \frac{3}{2x}\), understanding where to plot each point and how they relate to each other on this coordinate system is vital. Notice how, for inverse relationships like this one, the graph will show a hyperbolic curve, rather than a linear one, indicating that as one variable increases, the other decreases.