Question

Use the model \(y=\frac{12}{x} .\) Make a table of values for \(x=1,2,3,4,\) and \(5 .\) Sketch the graph.

Short answer

In the equation \(y=\frac{12}{x}\), using x values 1, 2, 3, 4 and 5 yield y values 12, 6, 4, 3 and 2.4 respectively and are plotted accordingly on the graph. The graph clearly shows an inverse relationship between x and y.

Step-by-step solution

Understanding the Equation

The equation given is \(y=\frac{12}{x}\). Here, \(y\) is inversely proportional to \(x\), which means when \(x\) increases, \(y\) decreases and vice versa.

Creating the Table

Substitute \(x=1,2,3,4,5\) into the equation. This will yield the respective \(y\) values. This can be done in the following progression: \(y_{1}=\frac{12}{1}=12\), \(y_{2}=\frac{12}{2}=6\), \(y_{3}=\frac{12}{3}=4\), \(y_{4}=\frac{12}{4}=3\), \(y_{5}=\frac{12}{5}=2.4\). Hence, the table becomes: (x=1,y=12), (x=2,y=6), (x=3,y=4), (x=4,y=3), (x=5,y=2.4).

Plotting the Graph

Each (x,y) pair corresponds to an individual point on plane. These points are plotted on the graph as follows: (1,12), (2,6), (3,4), (4,3), (5,2.4). After plotting, they will form a hyperbola that heads to infinity on the positive x side and to zero on the negative y side. The graph denotes an inverse relationship between x and y.

Key concepts

Inversely Proportional Relationships

Inversely proportional relationships, also known as inverse variation, occur when two variables are connected in such a way that as one variable increases, the other decreases. In the context of algebra, this is expressed as an equation like the one in our exercise, where we have the model \(y = \frac{12}{x}\). This equation means that for any value of \(x\), you can find the corresponding value of \(y\) by dividing the number 12 by the value of \(x\).

As an example, if you were to double the value of \(x\) from 1 to 2, the value of \(y\) would halve, going from 12 to 6. This type of relationship is fundamental in understanding how certain quantities in physics, such as pressure and volume in an ideal gas, interact with one another.

Plotting Graphs in Algebra

Plotting graphs in algebra is a vital skill for visualizing relationships between variables. For inverse variation, the graph will typically be a hyperbola, a curve open at both ends. When plotting the graph of \(y = \frac{12}{x}\), you'll notice that as \(x\) gets larger, \(y\) approaches zero, and as \(x\) approaches zero from the positive side, \(y\) will become very large, theoretically reaching infinity.

When plotting, it is important to carefully select a range of \(x\) values that will demonstrate the nature of the hyperbolic shape. As the points are plotted and connected, they should show the distinct inverse correlation where one variable increases as the other decreases, creating a curve that never touches either axis, indicating that neither \(x\) nor \(y\) will ever be zero within our inverse relationship.

Creating a Table of Values

Creating a table of values is an essential step before plotting the graph in algebra. It involves choosing specific values for one variable and calculating the corresponding values for the other variable. For the exercise involving \(y = \frac{12}{x}\), we start by selecting a set of values for \(x\) - typically, we choose positive integers that are easy to handle. In this case, we have \(x = 1, 2, 3, 4, 5\).

After choosing our values for \(x\), we solve for \(y\) by substituting each value into our equation. This results in a pair of \(x, y\) points that we can list in our table, providing a visual representation that assists in understanding how \(y\) changes as \(x\) varies. The table is an excellent tool for checking our work and ensuring that our plotted points will be accurate.