Problem 29 - Algebra 1: Concepts and Skills

Question

DIRECT OR INVERSE VARIATION Make a table of values for \(x=-4,-3\) \(-2,-1,1,2,3,\) and \(4 .\) Use the table to sketch the graph. State whether \(x\) and \(y\) vary directly or inversely. $$ y=3 x $$

Step-by-Step Solution

Answer

The values of y for the given x values in the equation \(y = 3x\) are mentioned in the table in step 1. The graph would be a straight line passing through the origin according to the plotted points. The relationship between x and y in the given equation is a Direct Variation, as increasing or decreasing x leads to directly increasing or decreasing y by a constant factor.

Step 1: Creating the Table

First, a table should be made to represent the values of y for each values of x as per the given equation. The table would be as follows:| \(x\) | \(y\) || --- | --- || -4 | -12 || -3 | -9 || -2 | -6 || -1 | -3 || 1 | 3 || 2 | 6 || 3 | 9 || 4 | 12 |

Step 2: Sketching the Graph

Using the table of values, the next step is to sketch the graph. Plot the given values of x and y as coordinates on the graph to get the points. Then, join these points in the order to get the graph. Since it's a direct variation, the graph will be a straight line passing through the origin.

Step 3: Direct or Inverse Variation

Finally, consider the nature of the graph and the equation. Since the equation is \(y = 3x\), it is obvious that y varies directly as x with a constant of variation equals to 3. Therefore, x and y vary directly, not inversely.

Key Concepts

inverse variation

Inverse variation describes a relationship where one variable increases as the other decreases. For two variables, let's say \(x\) and \(y\), they vary inversely if their product is a constant. This can be expressed mathematically as: \[ xy = k \] where \(k\) is the constant of variation. In simpler terms, if you multiply the values of \(x\) and \(y\), you always get the same number, \(k\). An everyday example could be the speed and time taken for a journey—the faster you go (increase speed), the shorter the time (decrease time) it takes. Inverse variations are graphically represented by hyperbolas, not straight lines.Identifying inverse variation can be a bit tricky at first. Remember:

• If the product of two variables is constant, the relationship is inverse.
• If the equation does not fit the form \(y = kx\) or results in a straight line through the origin, it is not inverse variation.
• Instead, if it fits \(xy = k\), you have an inverse relationship.

graphing linear equations

Graphing linear equations is a way to represent equations on a graph as straight lines. Linear equations, like \(y = 3x\), indicate that \(x\) and \(y\) have a direct linear relationship. This usually results in a straight line on the graph. Here are some steps to follow when graphing a linear equation:

• Select two or more values for \(x\), and use the equation to find the corresponding \(y\) values.
• Plot these \((x,y)\) pairs on a graph as points.
• Connect the points with a straight line. Extend the line, as needed, in both directions.

Linear graphs are characterized by their slope, which shows the rate of change of \(y\) with respect to \(x\). The slope is determined by the equation’s coefficient of \(x\), for example, \(3\) in \(y = 3x\).In practical terms, a positive slope means a line that slopes upwards, whereas a negative slope goes downwards. A slope of zero results in a horizontal line, showing no change in \(y\) as \(x\) increases.

table of values

A table of values is a great tool for organizing and calculating points to graph equations. Essentially, it is a list of different values for \(x\), and the corresponding \(y\) values calculated from an equation.Here’s how you can construct one:

• Choose a range for the \(x\) values. For linear equations, you can use positive and negative values to see the full line span.
• For each \(x\), substitute it into the equation to find the corresponding \(y\). This will provide a list of points that you can plot on a graph.
• List all \((x, y)\) pairs in a tabular format for easy reference and plotting.

Using a table of values helps ensure accuracy when graphing, as it provides clear points to plot. Once plotted, these points can be connected to form the proper shape dictated by the equation. This method is particularly useful for visualizing the function and understanding how \(x\) and \(y\) relate.

constant of variation

The constant of variation is the core fixed value that defines how two variables are related in a direct or inverse relationship.In direct variation, described by the equation \(y = kx\), \(k\) is the constant of variation. It indicates how much \(y\) changes when \(x\) changes. Think of it as the "multiplier" or "rate" that defines the relationship.Here are important points about it:

• If \(k\) is positive, \(y\) increases as \(x\) increases.
• If \(k\) is negative, \(y\) decreases as \(x\) increases.
• In direct relationships, \(k\) affects the steepness of the graph line.

For inverse variation, expressed as \(xy = k\), the constant \(k\) represents how the variables interact such that as one increases, the other decreases.Recognizing the constant of variation allows you to quickly understand the dynamics of the relationship between the variables and anticipate how changes in one will affect the other.